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Chapter 12 Resource Masters
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Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.
ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1
Spanish VersionHomework Practice Workbook 0-07-890853-1 978-0-07-890853-8
Answers for Workbooks The answers for Chapter 12 of these workbooks can be found in the back of this Chapter Resource Masters booklet.
StudentWorks PlusTM This CD-ROM includes the entire Student Edition text along with the English workbooks listed above.
TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.
Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 12 Test Form 2A and Form 2C.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.
Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 978-0-07-890521-6MHID: 0-07-890521-4
Printed in the United States of America.
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ContentsTeacher’s Guide to Using the Chapter 12Resource Masters .............................................iv
Chapter ResourcesChapter 12 Student-Built Glossary .................... 1Chapter 12 Anticipation Guide (English) ........... 3Chapter 12 Anticipation Guide (Spanish) .......... 4
Lesson 12-1Representations of Three-Dimensional FiguresStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10Graphing Calculator Activity ............................ 11
Lesson 12-2Surface Area of Prisms and CylindersStudy Guide and Intervention .......................... 12Skills Practice .................................................. 14
Practice ............................................................ 15Word Problem Practice ................................... 16Enrichment ...................................................... 17
Lesson 12-3Surface Area of Pyramids and ConesStudy Guide and Intervention .......................... 18Skills Practice .................................................. 20Practice ............................................................ 21Word Problem Practice ................................... 22Enrichment ...................................................... 23Spreadsheet Activity ........................................ 24
Lesson 12-4Volumes of Prisms and CylindersStudy Guide and Intervention .......................... 25Skills Practice .................................................. 27Practice ............................................................ 28Word Problem Practice ................................... 29Enrichment ...................................................... 30
Lesson 12-5Volumes of Pyramids and ConesStudy Guide and Intervention .......................... 31Skills Practice .................................................. 33Practice ............................................................ 34Word Problem Practice ................................... 35Enrichment ...................................................... 36
Lesson 12-6Surface Areas and Volumes of SpheresStudy Guide and Intervention .......................... 37Skills Practice .................................................. 39Practice ............................................................ 40Word Problem Practice ................................... 41Enrichment ...................................................... 42
Lesson 12-7Spherical GeometryStudy Guide and Intervention .......................... 43Skills Practice .................................................. 45Practice ............................................................ 46Word Problem Practice ................................... 47Enrichment ...................................................... 48
Lesson 12-8Congruent and Similar SolidsStudy Guide and Intervention .......................... 49Skills Practice .................................................. 51Practice ............................................................ 52Word Problem Practice ................................... 53Enrichment ...................................................... 54
AssessmentStudent Recording Sheet ................................ 55Rubric for Extended-Response ....................... 56Chapter 12 Quizzes 1 and 2 ........................... 57
Chapter 12 Quizzes 3 and 4 ........................... 58Chapter 12 Mid-Chapter Test .......................... 59Chapter 12 Vocabulary Test ........................... 60Chapter 12 Test, Form 1 ................................. 61Chapter 12 Test, Form 2A ............................... 63Chapter 12 Test, Form 2B ............................... 65Chapter 12 Test, Form 2C .............................. 67Chapter 12 Test, Form 2D .............................. 69Chapter 12 Test, Form 3 ................................. 71Chapter 12 Extended-Response Test ............. 73Standardized Test Practice ............................. 74
Answers ........................................... A1–A36
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Teacher’s Guide to Using the Chapter 12 Resource Masters
The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing on the TeacherWorks PlusTM CD-ROM.
Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 12–1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.
Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.
Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.
Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.
Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.
Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.
Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.
Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.
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Assessment OptionsThe assessment masters in the Chapter 12 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.
Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.
Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.
Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.
Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.
Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 12 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.
Leveled Chapter Tests• Form 1 contains multiple-choice
questions and is intended for use with approaching grade level students.
• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Form 3 is a free-response test for use with beyond grade level students.
All of the above mentioned tests include a free-response Bonus question.
Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.
Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.
Answers• The answers for the Anticipation Guide
and Lesson Resources are provided as reduced pages.
• Full-size answer keys are provided for the assessment masters.
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Chapter 12 1 Glencoe Geometry
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 12. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter.
Student-Built Glossary12
Vocabulary TermFound
on PageDefi nition/Description/Example
altitude
axis
congruent solids
cross section
Euclidean geometry
great circle
isometric view
lateral area
lateral edge
lateral face
(continued on the next page)
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Chapter 12 2 Glencoe Geometry
Vocabulary TermFound
on PageDefi nition/Description/Example
oblique cone
oblique cylinder
oblique prism
regular pyramid
right cone
right cylinder
right prism (PRIZ·uhm)
similar solids
slant height
spherical geometry
Student-Built Glossary (continued)12
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Chapter 12 3 Glencoe Geometry
Before you begin Chapter 12
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).
After you complete Chapter 12
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.
12 Anticipating GuideExtending Surface Area and Volume
Step 1
STEP 1A, D, or NS
StatementSTEP 2A or D
1. The shape of a horizontal cross section of a square pyramid is a triangle.
2. The lateral area of a prism is equal to the sum of the areas of each face.
3. The axis of an oblique cylinder is different than the height of the cylinder.
4. The slant height and height of a regular pyramid are the same.
5. The lateral area of a cone equals the product of π, the radius, and the height of the cone.
6. The volume of a right cylinder with radius r and height h is πr2h.
7. The volume of a pyramid or a cone is found by multiplying the area of the base by the height.
8. To find the surface area of a sphere with radius r, multiply πr2 by 4.
9. All postulates and properties of Euclidean geometry are true in spherical geometry.
10. All spheres and all cubes are similar solids.
Step 2
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NOMBRE FECHA PERÍODO
PDF 2nd
Antes de comenzar el Capítulo 12
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a).
Después de completar el Capítulo 12
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.
12 Ejercicios PreparationsExtiende el Área de Superficie y volumen
Paso 1
PASO 1A, D o NS
EnunciadoPASO 2A o D
1. La forma de un corte transversal horizontal de una pirámide cuadrada es un triángulo.
2. El área lateral de un prisma es igual a la suma de las áreas de cada cara.
3. El eje de un cilindro oblicuo es diferente a la altura del cilindro.
4. La altura oblicua y la altura de una pirámide regular son las mismas.
5. El área lateral de un cono es igual al producto de π, el radio, por la altura del cono.
6. El volumen de un cilindro recto con radio r y altura h es πr2h.
7. El volumen de una pirámide o un cono se calcula multiplicando el área de la base por la altura.
8. Para calcular el área de superficie de una esfera con radio r, multiplica πr2 por 4.
9. Todos los postulados y propiedades de la geometría euclidiana son verdaderos en geometría esférica.
10. Todas las esferas y todos los cubos son sólidos semejantes.
Paso 2
Capítulo 12 4 Geometría de Glencoe
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Chapter 12 5 Glencoe Geometry
Draw Isometric Views Isometric dot paper can be used to draw isometric views, or corner views, of a three-dimensional object on two-dimensional paper.
Use isometric dot paper to sketch a triangular prism 3 units high, with two sides of the base that are 3 unitslong and 4 units long.Step 1 Draw
−−
AB at 3 units and draw −−
AC at 4 units.Step 2 Draw
−−−
AD , −−−
BE , and −−
CF , each at 3 units.Step 3 Draw
−−−
BC and �DEF.
Use isometric dot paper and the orthographic drawing to sketch a solid. • The top view indicates two columns. • The right and left views indicate that the height of figure is
three blocks. • The front view indicates that the columns have heights 2 and 3 blocks.
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of each column.
ExercisesSketch each solid using isometric dot paper.
1. cube with 4 units on each side 2. rectangular prism 1 unit high, 5 units long, and 4 units wide
Use isometric dot paper and each orthographic drawing to sketch a solid.
3.
top view left view front view right view
4.
top view left view front view right view
Study Guide and InterventionRepresentations of Three-Dimensional Figures
12-1
Example 1
A
BC
D
E F
Example 2
top view left view front view right view
object
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Chapter 12 6 Glencoe Geometry
Cross Sections The intersection of a solid and a plane is called a cross section of the solid. The shape of a cross section depends upon the angle of the plane.
There are several interesting shapes that are cross sections of a cone. Determine the shape resulting from each cross section of the cone.
a. If the plane is parallel to the base of the cone, then the resulting cross section will be a circle.
b. If the plane cuts through the cone perpendicular to the base and through the center of the cone, then the resulting cross section will be a triangle.
c. If the plane cuts across the entire cone, thenthe resulting cross section will be an ellipse.
ExercisesDescribe each cross section.
1. 2. 3.
Study Guide and Intervention (continued)
Representations of Three-Dimensional Figures
12-1
Example
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Chapter 12 7 Glencoe Geometry
12-1
Use isometric dot paper to sketch each prism.
1. cube 2 units on each edge 2. rectangular prism 2 units high, 5 units long, and 2 units wide
Use isometric dot paper and each orthographic drawing to sketch a solid.
3. 4.
Describe each cross section.
5. 6.
7. 8.
Skills PracticeRepresentations of Three-Dimensional Figures
top view left view front view right viewtop view left view front view right view
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Chapter 12 8 Glencoe Geometry
Use isometric dot paper to sketch each prism.
1. rectangular prism 3 units high, 2. triangular prism 3 units high, whose bases 3 units long, and 2 units wide are right triangles with legs 2 units and 4 units long
Use isometric dot paper and each orthographic drawing to sketch a solid.
3.
top view left view front view right view
4.
top view left view front view right view
Sketch the cross section from a vertical slice of each figure.
5. 6.
7. SPHERES Consider the sphere in Exercise 5. Based on the cross section resulting from a horizontal and a vertical slice of the sphere, make a conjecture about all spherical cross sections.
8. MINERALS Pyrite, also known as fool’s gold, can form crystals that are perfect cubes. Suppose a gemologist wants to cut a cube of pyrite to get a square and a rectanglar face. What cuts should be made to get each of the shapes? Illustrate your answers.
PracticeRepresentations of Three-Dimensional Figures
12-1
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Chapter 12 9 Glencoe Geometry
1. LABELS Jamal removes the label from a cylindrical soup can to earn points for his school. Sketch the shape of the label.
2. BLOCKS Margot’s three-year-old son made the magnetic block sculpture shown below in corner view.
Draw the right view of the sculpture.
3. CUBES Nathan marks the midpoints of three edges of a cube as shown.He then slices the cube along a plane that contains these three points. Describe the resulting cross section.
4. ENGINEERING Stephanie needs an object whose top view is a circle and whose left and front views are squares. Describe an object that will satisfy these conditions.
5. DESK SUPPORTS The figure shows the support for a desk.
a. Draw the top view.
b. Draw the front view.
c. Draw the right view.
Word Problem PracticeRepresentations of Three-Dimensional Figures
12-1
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Chapter 12 10 Glencoe Geometry
Drawing Solids on Isometric Dot PaperIsometric dot paper is helpful for drawing solids. Remember to use dashed lines for hidden edges.
For each solid shown, draw another solid whose dimensions are twice as large.
1. 2.
3. 4.
5. 6.
Enrichment12-1
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Chapter 12 11 Glencoe Geometry
The science of perspective drawing studies how to draw a three-dimensional object on a two-dimensional page. This science became highly refined during the Renaissance with the work of artists such as Albrecht Dürer and Leonardo da Vinci.
Today, computers are often used to make perspective drawings, particularly elaborate graphics used in television and movies. The three-dimensional coordinates of objects are figured. Then algebra is used to transform these into two-dimensional coordinates. The graph of these new coordinates is called a projection.
The formulas below will draw one type of projection in which the y-axis is drawn horizontally, the z-axis vertically, and the x-axis at an angle of a˚ with the y-axis. If the three-dimensional coordinates of a point are (x, y, z), then the projection coordinates (X, Y) are given by
X = x(-cos a) + y and Y = x(-sin a) + z.Although this type of projection gives a fairly good perspective drawing, it does distort some lengths.
1. The drawing with the coordinates given below is a cube. A(5, 0, 5), B(5, 5, 5), C(5, 5, 0), D(5, 0, 0),
E(0, 0, 5), F(0, 5, 5), G(0, 5, 0), H(0, 0, 0) Use the formulas above to find the projection coordinates of each
point, using a = 45. Round projection coordinates to the nearest integer. Graph the cube on a graphing calculator. Make a sketch of the display.
A'(__, __) B'(__, __) C′(__, __) D′(__, __) E'(__, __) F'(__, __) G′(__, __) H′(__, __)
2. The points A(10, 2, 0), B(10, 10, 0), C(2, 10, 0), and D(3, 3, 4) are vertices of a pyramid. Find the projection coordinates, using a = 25. Round coordinates to the nearest integer. Then graph the pyramid on a graphing calculator by drawing
−−−
A′B′ , −−−
B′C′ , −−−
C′D′ , −−−
D′A′ , and
−−−
D′B′ . Make a sketch of the display. A′(__, __) B′(__, __) C′(__, __) D′(__, __)
Graphing Calculator ActivityPerspective Drawings
12-1
E F
A B
D
H
C
G
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Chapter 12 12 Glencoe Geometry
Study Guide and InterventionSurface Areas of Prisms and Cylinders
12-2
Lateral and Surface Areas of Prisms In a solid figure, faces that are not bases are lateral faces. The lateral area is the sum of the area of the lateral faces. The surface area is the sum of the lateral area and the area of the bases.
Lateral Area
of a Prism
If a prism has a lateral area of L square units, a height of h units,
and each base has a perimeter of P units, then L = Ph.
Surface Area
of a Prism
If a prism has a surface area of S square units, a lateral area of L
square units, and each base has an area of B square units, then
S = L + 2B or S = Ph + 2B
Find the lateral and surface area of the regular pentagonal prism above if each base has a perimeter of 75 centimeters and the height is 10 centimeters.
L = Ph Lateral area of a prism
= 75(10) P = 75, h = 10
= 750 Multiply.
The lateral area is 750 square centimeters and the surface area is about 1524.2 square centimeters.
ExercisesFind the lateral area and surface area of each prism. Round to the nearest tenth if necessary.
1.
4 m
3 m10 m
2.
15 in.
10 in.
8 in.
3.
6 in.18 in.
4.
20 cm
10 cm 10 cm
12 cm8 cm9 cm
5.
4 in.
4 in.
12 in.
6.
4 m16 m
Example
pentagonal prism
altitude
lateraledge lateral
face
S = L + 2B = 750 + 2 ( 1 −
2 aP)
= 750 + ( 7.5 − tan 36°
) (75)
≈ 1524.2tan 36° = 7.5 − a
a = 7.5 − tan 36°
a
15 cm
36°
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Chapter 12 13 Glencoe Geometry
Lateral and Surface Areas of Cylinders A cylinder is a solid with bases that are congruent circles lying in parallel planes. The axis of a cylinder is the segment with endpoints at the centers of these circles. For a right cylinder, the axis is also the altitude of the cylinder.
Lateral Area
of a Cylinder
If a cylinder has a lateral area of L square units, a height of h units, and a base
has a radius of r units, then L = 2πrh.
Surface Area
of a Cylinder
If a cylinder has a surface area of S square units, a height of h units, and a
base has a radius of r units, then S = L + 2B or 2πrh + 2πr2.
Find the lateral and surface area of the cylinder. Round to the nearest tenth.If d = 12 cm, then r = 6 cm.L = 2πrh Lateral area of a cylinder
= 2π(6)(14) r = 6, h = 14
≈ 527.8 Use a calculator.
S = 2πrh + 2πr2 Surface area of a cylinder
≈ 527.8 + 2π(6)2 2πrh ≈ 527.8, r = 6
≈ 754.0 Use a calculator.
The lateral area is about 527.8 square centimeters and the surface area is about 754.0 square centimeters.
ExercisesFind the lateral area and surface area of each cylinder. Round to the nearest tenth. 1.
12 cm
4 cm 2.
6 in.10 in.
3.
6 cm
3 cm
3 cm
4.
20 cm
8 cm
5.
12 m
4 m
6. 2 m
1 m
radius of baseaxis
base
baseheight
Example
14 cm
12 cm
12-2 Study Guide and Intervention (continued)
Surface Areas of Prisms and Cylinders
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Chapter 12 14 Glencoe Geometry
Find the lateral area and surface area of each prism. Round to the nearest tenth if necessary.
1. 2.
3. 4.
Find the lateral area and surface area of each cylinder. Round to the nearest tenth.
5. 6.
7. 8.
12 yd
12 yd
10 yd8 m
6 m
12 m
10 in.5 in.
6 in.
8 in.
9 cm
9 cm
7.8 cm9 cm
12 cm
Skills PracticeSurface Areas of Prisms and Cylinders
12-2
8 in.
12 in.
2 yd
3 yd
12 in.
10 in.
2 m
2 m
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Chapter 12 15 Glencoe Geometry
12-2 PracticeSurface Areas of Prisms and Cylinders
Find the lateral and surface area of each prism. Round to the nearest tenth if necessary. 1.
15 cm
15 cm
32 cm
2.
8 ft
10 ft5 ft
3.
2 m11 m
4.
4 yd
4 yd
9.5 yd
5 yd
Find the lateral area and surface area of each cylinder. Round to the nearest tenth.
5. 5 ft
7 ft
6. 4 m
8.5 m
7. 19 in.
17 in.
8.
12 m30 m
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Chapter 12 16 Glencoe Geometry
1. LOGOS The Z company specializes in caring for zebras. They want to make a 3-dimensional “Z” to put in front of their company headquarters. The “Z” is 15 inches thick and the perimeter of the base is 390 inches.
15"
What is the lateral surface area of this “Z”?
2. STAIRWELLS Management decides to enclose stairs connecting the first and second floors of a parking garage in a stairwell shaped like an oblique rectangular prism.
16 ft
20 ft
15 ft
9 ft
What is the lateral surface area of the stairwell?
3. CAKES A cake is a rectangular prism with height 4 inches and base 12 inches by 15 inches. Wallace wants to apply frosting to the sides and the top of the cake. What is the surface area of the part of the cake that will have frosting?
4. EXHAUST PIPES An exhaust pipe is shaped like a cylinder with a height of 50 inches and a radius of 2 inches. What is the lateral surface area of the exhaust pipe? Round your answer to the nearest hundredth.
5. TOWERS A circular tower is made by placing one cylinder on top of another. Both cylinders have a height of 18 inches. The top cylinder has a radius of 18 inches and the bottom cylinder has a radius of 36 inches.
18 in.
18 in.
a. What is the total surface area of the tower? Round your answer to the nearest hundredth.
b. Another tower is constructed by placing the original tower on top of another cylinder with a height of 18 inches and a radius of 54 inches. What is the total surface area of the new tower? Round your answer to the nearest hundredth.
12-2 Word Problem PracticeSurface Areas of Prisms and Cylinders
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Chapter 12 17 Glencoe Geometry
Enrichment12-2
Minimizing Cost in ManufacturingSuppose that a manufacturer wants to make a can that has a volume of 40 cubic inches. The cost to make the can is 3 cents per square inch for the top and bottom and 1 cent per square inch for the side.
1. Write the value of h in terms of r, given v = πr2h.
2. Write a formula for the cost in terms of r.
3. Use a graphing calculator to graph the formula, letting Y1 represent the cost and X represent r. Use the graph to estimate the point at which the cost is minimized.
4. Repeat the procedure using 2 cents per square inch for the top and bottom and 4 cents per square inch for the top and bottom.
5. What would you expect to happen as the cost of the top and bottom increases?
6. Compute the table for the cost value given. What happens to the height of the can as the cost of the top and bottom increases?
Cost Top Cost Minimum
& Bottom Cylinder h
2 cents 1 cent
3 cents 1 cent
4 cents 1 cent
5 cents 1 cent
6 cents 1 cent
r
h
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Chapter 12 18 Glencoe Geometry
12-3 Study Guide and InterventionSurface Areas of Pyramids and Cones
Lateral and Surface Areas of Pyramids A pyramid is a solid with a polygon base. The lateral faces intersect in a common point known as the vertex. The altitude is the segment from the vertex that is perpendicular to the base. For a regular pyramid, the base is a regular polygon and the altitude has an endpoint at the center of the base. All the lateral edges are congruent and all the lateral faces are congruent isosceles triangles. The height of each lateral face is called the slant height.
Lateral Area of
a Regular Pyramid
The lateral area L of a regular pyramid is L = 1 − 2 Pℓ, where ℓ
is the slant height and P is the perimeter of the base.
Surface Area of
a Regular Pyramid
The surface area S of a regular pyramid is S = 1 − 2 Pℓ + B,
where ℓ is the slant height, P is the perimeter of the base,
and B is the area of the base.
For the regular square pyramid above, find the lateral area and surface area if the length of a side of the base is 12 centimeters and the height is 8 centimeters. Round to the nearest tenth if necessary.
Find the slant height.ℓ2 = 62 + 82 Pythagorean Theorem
ℓ2 = 100 Simplify.
ℓ = 10 Take the positive square root of each side.
The lateral area is 240 square centimeters, and the surface area is 384 square centimeters.
ExercisesFind the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.
1.
15 cm
20 cm
2.
45°
8 ft
3.
60°
10 cm 4.
6 in.8.7 in. 15 in.
lateral edge
baseslant height height
Example
L = 1 − 2 Pℓ Lateral area of a regular pyramid
= 1 − 2 (48)(10) P = 4 � 12 or 48, ℓ = 10
= 240 Simplify.
S = 1 − 2 Pℓ + B Surface area of a regular pyramid
= 240 + 144 1 − 2 Pℓ = 240, B = 12 · 12 or 144
= 384
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Chapter 12 19 Glencoe Geometry
12-3 Study Guide and Intervention (continued)
Surface Areas of Pyramids and Cones
Lateral and Surface Areas of Cones A cone has a circular base and a vertex. The axis of the cone is the segment with endpoints at the vertex and the center of the base. If the axis is also the altitude, then the cone is a right cone. If the axis is not the altitude, then the cone is an oblique cone.
Lateral Area of
a Cone
The lateral area L of a right circular cone is L = πr�, where r is
the radius and � is the slant height.
Surface Area of
a Cone
The surface area S of a right cone is S = πr� + πr2, where r is
the radius and � is the slant height.
For the right cone above, find the lateral area and surface area if the radius is 6 centimeters and the height is 8 centimeters. Round to the nearest tenth if necessary.
Find the slant height.ℓ2 = 62 + 82 Pythagorean Theorem
ℓ2 = 100 Simplify.
ℓ = 10 Take the positive square root of each side.
The lateral area is about 188.5 square centimeters and the surface area is about 301.6 square centimeters.
ExercisesFind the lateral area and surface area of each cone. Round to the nearest tenth if necessary.
1.
9 cm
12 cm 2.
5 ft
30°
3. 12 cm
13 cm
4.
4 in.
45°
axis
base base�
slant height
right coneoblique cone
altitudeV V
Example
L = πrℓ Lateral area of a right cone
= π(6)(10) r = 6, ℓ = 10
≈ 188.5 Simplify.
S = πrℓ + πr2 Surface area of a right cone
≈ 188.5 + π(62) πrℓ ≈ 188.5, r = 6
≈ 301.6 Simplify.
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Chapter 12 20 Glencoe Geometry
12-3 Skills PracticeSurface Areas of Pyramids and Cones
Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.
1. 2.
3. 4.
Find the lateral area and surface area of each cone. Round to the nearest tenth.
5. 6.
7. 8.
4 cm
7 cm20 in.
8 in.
9 m
10 m 14 ft
12 ft
14 m
5 m 10 ft
25 ft
8 in.
21 in.
17 mm
9 mm
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Chapter 12 21 Glencoe Geometry
12-3 Practice Surface Areas of Pyramids and Cones
Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.
1.
9 yd
10 yd
2. 12 m
7 m
3.
13 ft
5 ft
4.
8 cm
2.5 cm
Find the lateral area and surface area of each cone. Round to the nearest tenth if necessary.
5.
5 m4 m
6. 7 cm
21 cm
7. Find the surface area of a cone if the height is 14 centimeters and the slant height is 16.4 centimeters.
8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.
9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the pyramid has sides of 0.5 meter and the slant height of the roof is 1.9 meters, find the area of the roof.
10. HATS Cuong bought a conical hat on a recent trip to central Vietnam. The basic frame of the hat is 16 hoops of bamboo that gradually diminish in size. The hat is covered in palm leaves. If the hat has a diameter of 50 centimeters and a slant height of 32 centimeters, what is the lateral area of the conical hat?
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Chapter 12 22 Glencoe Geometry
12-3 Word Problem PracticeSurface Areas of Pyramids and Cones
1. PAPER MODELS Patrick is making a paper model of a castle.Part of the model involves cutting out the net shown and folding it into a pyramid. The pyramid has a square base. What is the lateral surface area of the resulting pyramid?
2. TETRAHEDRON Sung Li builds a paper model of a regular tetrahedron, a pyramid with an equilateral triangle for the base and three equilateral triangles for the lateral faces. One of the faces of the tetrahedron has an area of 17 square inches. What is the total surface area of the tetrahedron?
3. PAPERWEIGHTS Daphne uses a paperweight shaped like a pyramid with a regular hexagon for a base. The side length of the regular hexagon is 1 inch. The altitude of the pyramid is 2 inches.
What is the lateral surface area of this pyramid? Round your answers to the nearest hundredth.
4. SPRAY PAINT A can of spray paint shoots out paint in a cone shaped mist. The lateral surface area of the cone is 65π square inches when the can is held 12 inches from a canvas. What is the area of the part of the canvas that gets sprayed with paint? Round your answer to the nearest hundredth.
5. MEGAPHONES A megaphone is formed by taking a cone with a radius of 20 centimeters and an altitude of 60 centimeters and cutting off the tip. The cut is made along a plane that is perpendicular to the axis of the cone and intersects the axis 12 centimeters from the vertex. Round your answers to the nearest hundredth.
a. What is the lateral surface area of the original cone?
b. What is the lateral surface area of the tip that is removed?
c. What is the lateral surface area of the megaphone?
20 cm20 cm15 cm
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Chapter 12 23 Glencoe Geometry
Cone Patterns
The pattern at the right is made from a circle. It can be folded to make a cone.
1. Measure the radius of the circle to the nearest centimeter.
2. The pattern is what fraction of the complete circle?
3. What is the circumference of the complete circle?
4. How long is the circular arc that is the outside of the pattern?
5. Cut out the pattern and tape it together to form a cone.
6. Measure the diameter of the circular base of the cone.
7. What is the circumference of the base of the cone?
8. What is the slant height of the cone?
9. Use the Pythagorean Theorem to calculate the height of the cone. Use a decimal approximation. Check your calculation by measuring the height with a metric ruler.
10. Find the lateral area.
11. Find the total surface area.
Make a paper pattern for each cone with the given measurements. Then cut the pattern out and make the cone. Find the measurements.
12.
120°6 cm
13.
20 cm
diameter of base = diameter of base =
lateral area = lateral area =
height of cone = height of cone =(to nearest tenth of a centimeter) (to nearest tenth of a centimeter)
Enrichment12-3
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Chapter 12 24 Glencoe Geometry
12-3
You can use a spreadsheet to determine the surface area of a cone.
Lucy wants to wrap a Mother’s Day gift. The gift she has bought for her mother is in a conical box that has a slant height of 6 inches and has a radius of 3 inches. She must determine the surface area of the box to determine how much wrapping paper to buy. Use a spreadsheet to determine the surface area of the box. Round to the nearest tenth.
Step 1 Use cell A1 for the radius of the cone and cell B1 for the height.
Step 2 In cell C1, enter an equals sign followed by PI()*A1*B1 + PI()*A1^2. Then press ENTER. This will return the surface area of the cone.
The surface area of the conical box is 84.8 in2 to the nearest tenth.
Use a spreadsheet to determine the surface area of a cone that has a radius of 2.5 centimeters and a slant height of 5.2 centimeters. Round to the nearest tenth.
Step 1 Use cell A2 for the radius of the cone and cell B2 for the slant height.
Step 2 Click on the bottom right corner of cell C1 and drag it to C2. This returns the surface area of the cone.
The surface area of the cone is 60.5 cm2 to the nearest tenth.
ExercisesUse a spreadsheet to find the surface area of each cone with the given dimensions. Round to the nearest tenth.
1. r = 12 m, � = 2.3 m 2. r = 6 m, � = 2 m
3. r = 3 in., � = 7 in. 4. r = 5 in., � = 11 in.
5. r = 1 ft, � = 3 ft 6. r = 3 ft, � = 1.5 ft
7. r = 10 mm, � = 20 mm 8. r = 1.5 mm, � = 4.5 mm
9. r = 6.2 cm, � = 1.2 cm 10. r = 10 cm, � = 15 cm
11. r = 10 m, � = 2 m 12. r = 11 m, � = 13 m
Spreadsheet ActivitySurface Areas of Cones
Example 1
Example 2
A12
B C
Sheet 1
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Chapter 12 25 Glencoe Geometry
Study Guide and InterventionVolumes of Prisms and Cylinders
Volumes of Prisms The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume is measured in units such as cubic feet, cubic yards, or cubic meters. One cubic unit is the volume of a cube that measures one unit on each edge.
Volume
of a Prism
If a prism has a volume of V cubic units, a height of h units,
and each base has an area of B square units, then V = Bh.
Find the volume of the prism.
7 cm3 cm
4 cm
V = Bh Volume of a prism
= (7)(3)(4) B = (7)(3), h = 4
= 84 Multiply.
The volume of the prism is 84 cubic centimeters.
Find the volume of the prism if the area of each base is 6.3 square feet.
3.5 ft
base
V = Bh Volume of a prism
= (6.3)(3.5) B = 6.3, h = 3.5
= 22.05 Multiply.The volume is 22.05 cubic feet.
Exercises
Find the volume of each prism.
1.
8 ft
8 ft
8 ft
2.
3 cm
4 cm
1.5 cm
3.
30°15 ft
12 ft 4.
10 ft15 ft
12 ft
5.
4 cm
6 cm
2 cm
1.5 cm
6.
7 yd4 yd
3 yd
12-4
Example 1 Example 2
cubic foot cubic yard27 cubic feet = 1 cubic yard
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Chapter 12 26 Glencoe Geometry
Study Guide and Intervention (continued)
Volumes of Prisms and Cylinders
Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. When a solid is not a right solid, use Cavalieri’s Priniciple to find the volume. The principle states that if two solids have the same height and the same cross sectional area at every level, then they have the same volume.
Find the volume of the cylinder.
4 cm
3 cm
V = πr2h Volume of a cylinder
= π(3)2(4) r = 3, h = 4
≈ 113.1 Simplify.
The volume is about 113.1 cubic centimeters.
Find the volume of the oblique cylinder.
8 in.
13 in.
5 in.
h
Use the Pythagorean Theorem to find the height of the cylinder.h2 + 52 = 132 Pythagorean Theorem
h2 = 144 Simplify.
h = 12 Take the positive square root of each side.
V = πr2h Volume of a cylinder
= π(4)2(12) r = 4, h = 12
≈ 603.2 Simplify.
The Volume is about 603.2 cubic inches.ExercisesFind the volume of each cylinder. Round to the nearest tenth.
1. 2 ft
1 ft
2. 18 cm
2 cm
3.
12 ft1.5 ft
4. 20 ft
20 ft
5.
10 cm
13 cm
6.
1 yd4 yd
12-4
Volume of
a Cylinder
If a cylinder has a volume of V cubic units, a height of h units,
and the bases have a radius of r units, then V = πr 2h.
Example 1 Example 2
r
h
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Chapter 12 27 Glencoe Geometry
12-4
Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.
1.
18 cm
16 cm
8 cm 2.
6 ft
8 ft
2 ft
3.
3 m
5 m
13 m
4.
16 in. 22 in.
34 in.
5.
15 mm23 mm
6. 6 yd
10 yd
Find the volume of each oblique prism or cylinder. Round to the nearest tenth if necessary.
7. 8.
5 in.
3 in.
Skills PracticeVolumes of Prisms and Cylinders
17 cm
18 cm
4 cm
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Chapter 12 28 Glencoe Geometry
PracticeVolumes of Prisms and Cylinders
Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.
1.
17 m10 m
26 m 2.
5 in.
5 in.
5 in.
9 in.
3.
16 mm 17.5 mm
4. 7 ft 25 ft
5.
13 yd
20 yd
10 yd 6.
30 cm
8 cm
7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25 1 −
2 inches and a radius of 21 inches.
a. What is the volume of the aquarium in cubic feet?
b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium hold?
c. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in the aquarium to the nearest five pounds?
12-4
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Chapter 12 29 Glencoe Geometry
Word Problem PracticeVolumes of Prisms and Cylinders
1. TRASH CANS The Meyer family uses a kitchen trash can shaped like a cylinder. It has a height of 18 inches and a base diameter of 12 inches.What is the volume of the trash can? Round your answer to the nearest tenth of a cubic inch.
2. BENCH Inside a lobby, there is a piece of furniture for sitting. The furniture is shaped like a simple block with a square base 6 feet on each side and a height of 1 3−
5 feet.
6 ft6 ft
1 ft35
What is the volume of the seat?
3. FRAMES Margaret makes a square frame out of four pieces of wood. Each piece of wood is a rectangular prism with a length of 40 centimeters, a height of 4 centimeters, and a depth of 6 centimeters. What is the total volume of the wood used in the frame?
4. PENCIL GRIPS A pencil grip is shaped like a triangular prism with a cylinder removed from the middle. The base of the prism is a right isosceles triangle with leg lengths of 2 centimeters. The diameter of the base of the removed cylinder is 1 centimeter. The heights of the prism and the cylinder are the same, and equal to 4 centimeters.
What is the exact volume of the pencil grip?
5. TUNNELS Construction workers are digging a tunnel through a mountain. The space inside the tunnel is going to be shaped like a rectangular prism. The mouth of the tunnel will be a rectangle 20 feet high and 50 feet wide and the length of the tunnel will be 900 feet.
a. What will the volume of the tunnel be?
b. If instead of a rectangular shape, the tunnel had a semicircular shape with a 50-foot diameter, what would be its volume? Round your answer to the nearest cubic foot.
18 in.
12 in.
12-4
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Chapter 12 30 Glencoe Geometry
Visible Surface AreaUse paper, scissors, and tape to make five cubes that have one-inch edges. Arrange the cubes to form each shape shown. Then find the volume and the visible surface area. In other words, do not include the area of surface covered by other cubes or by the table or desk.
1. 2.
volume = volume =
visible surface area = visible surface area =
3. 4. 5.
volume = volume = volume =
visible surface area = visible surface area = visible surface area =
6. Find the volume and the visible surface area of the figure at the right.
volume =
visible surface area =
4 in.
4 in.
3 in.
8 in.
3 in.
5 in.
5 in.
3 in.
3 in.
12-4 Enrichment
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Chapter 12 31 Glencoe Geometry
Study Guide and InterventionVolumes of Pyramids and Cones
Volumes of Pyramids This figure shows a prism and a pyramid that have the same base and the same height. It is clear that the volume of the pyramid is less than the volume of the prism. More specifically, the volume of the pyramid is one-third of the volume of the prism.
Find the volume of the square pyramid.
V = 1 − 3 Bh Volume of a pyramid
= 1 − 3 (8)(8)10 B = (8)(8), h = 10
≈ 213.3 Multiply.
The volume is about 213.3 cubic feet.
ExercisesFind the volume of each pyramid. Round to the nearest tenth if necessary.
1.
12 ft
8 ft
10 ft 2.
10 ft
6 ft15 ft
3.
4 cm8 cm
12 cm 4.
18 ft
regularhexagon 6 ft
5.
15 in.
15 in.
16 in. 6. 6 yd
8 yd
5 yd
8 ft
8 ft
10 ft
12-5
Volume of
a Pyramid
If a pyramid has a volume of V cubic units, a height of h units,
and a base with an area of B square units, then V = 1 − 3 Bh.
Example
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Chapter 12 32 Glencoe Geometry
Volumes of Cones For a cone, the volume is one-third the product of the height and the area of the base. The base of a cone is a circle, so the area of the base is πr2.
Find the volume of the cone.
V = 1 − 3 πr2h Volume of a cone
= 1 − 3 π(5)212 r = 5, h = 12
≈ 314.2 Simplify.
The volume of the cone is about 314.2 cubic centimeters.
ExercisesFind the volume of each cone. Round to the nearest tenth.
1.
6 cm10 cm
2. 8 ft
10 ft
3.
30 in.
12 in.
4. 45°
18 yd
20 yd
5. 26 ft
20 ft
6.
16 cm
45°
12 cm
5 cm
Study Guide and Intervention (continued)
Volumes of Pyramids and Cones
12-5
Volume of
a Cone
If a cone has a volume of V cubic units, a height of h units,
and the bases have a radius of r units, then V = 1 − 3 πr2h.
Example
r
h
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Chapter 12 33 Glencoe Geometry
12-5 Skills PracticeVolumes of Pyramids and Cones
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary. 1. 2.
3. 4.
5. 6.
Find the volume of each oblique pyramid or cone. Round to the nearest tenth if necessary.
7. 8.
5 ft5 ft
8 ft
4 cm7 cm
8 cm
8 in.10 in.
14 in.25 m
12 m
25 yd
14 yd66°
18 mm
4 ft4 ft
6 ft12 cm
6 cm
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Chapter 12 34 Glencoe Geometry
PracticeVolumes of Pyramids and Cones
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
1.
9.2 yd9.2 yd
13 yd
2.
12.5 cm25 cm
23 cm
3.
19 ft
9 ft 4.
52°
12 mm
5.
6 in.6 in.
11 in.
6.
37 ft11 ft
7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in diameter and the height of the shed is 3.8 meters. What is the volume of the shed?
8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the 27th century B.C. In its original state, it stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid.
12-5
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Chapter 12 35 Glencoe Geometry
1. ICE CREAM DISHES The part of a dish designed for ice cream is shaped like an upside-down cone. The base of the cone has a radius of 2 inches and the height is 1.2 inches.
What is the volume of the cone? Round your answer to the nearest hundredth.
2. GREENHOUSES A greenhouse has the shape of a square pyramid. The base has a side length of 30 yards. The height of the greenhouse is 18 yards.
18
yd
30 yd
What is the volume of the greenhouse?
3. TEEPEE Caitlyn made a teepee for a class project. Her teepee had a diameter of 6 feet. The angle the side of the teepee made with the ground was 65°.
65˚
What was the volume of the teepee? Round your answer to the nearest hundredth.
4. SCULPTING A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the base of the cone and cylinder is 2 feet.
What is the volume of the stone that the sculptor must remove? Round your answer to the nearest hundredth.
5. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the top square is 12 feet and the side length of the bottom square is 16 feet. The height of the stage is 3 feet.
12 feet
16 feet
3 feet
a. What is the volume of the entire square pyramid that the stage is part of?
b. What is the volume of the top of the pyramid that is removed to get the stage?
c. What is the volume of the stage?
Word Problem PracticeVolumes of Pyramids and Cones
12-5
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Chapter 12 36 Glencoe Geometry
Enrichment
FrustumsA frustum is a figure formed when a plane intersects a pyramid or cone so that the plane is parallel to the solid’s base. The frustum is the part of the solid between the plane and the base. To find the volume of a frustum, the areas of both bases must be calculated and used in the formula.
V = 1 − 3 h(B1 + B2 + √ �� B1B2 ),
where h = height (perpendicular distance between the bases),B1 = area of top base, and B2 = area of bottom base.
Describe the shape of the bases of each frustum. Then find the volume. Round to the nearest tenth.
1. 13 cm
6 cm
9 cm
5 cm
19.5 cm
2.
7.5 in.
4.5 in.
3 in.
3.
8 m
6 m
12 m
4.5 m2.25 m
3 m
5 m
4.
12 ft13 ft
7 ft
12-5
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Chapter 12 37 Glencoe Geometry
Study Guide and InterventionSurface Areas and Volumes of Spheres
Surface Areas of Spheres You can think of the surface area of a sphere as the total area of all of the nonoverlapping strips it would take to cover the sphere. If r is the radius of the sphere, then the area of a great circle of the sphere is πr2. The total surface area of the sphere is four times the area of a great circle.
Find the surface area of a sphere to the nearest tenth if the radius of the sphere is 6 centimeters.
S = 4πr2 Surface area of a sphere
= 4π(6)2 r = 6
≈ 452.4 Simplify.
The surface area is 452.4 square centimeters.
ExercisesFind the surface area of each sphere or hemisphere. Round to the nearest tenth.
1. 5 m
2.
7 in
3.
3 ft
4.
9 cm
5. sphere: circumference of great circle = π cm
6. hemisphere: area of great circle ≈ 4π ft2
r
6 cm
Surface Area
of a SphereIf a sphere has a surface area of S square units and a radius of r units, then S = 4πr2.
Example
12-6
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Chapter 12 38 Glencoe Geometry
Study Guide and Intervention (continued)
Surface Areas and Volumes of Spheres
12-6
Volumes of Spheres A sphere has one basic measurement, the length of its radius. If you know the length of the radius of a sphere, you can calculate its volume.
Find the volume of a sphere with radius 8 centimeters.
V = 4 − 3 πr3 Volume of a sphere
= 4 − 3 π (8)3 r = 8
≈ 2144.7 Simplify.
The volume is about 2144.7 cubic centimeters.
ExercisesFind the volume of each sphere or hemisphere. Round to the nearest tenth.
1.
5 ft
2. 6 in. 3.
16 in.
4. hemisphere: radius 5 in.
5. sphere: circumference of great circle ≈ 25 ft
6. hemisphere: area of great circle ≈ 50 m2
r
8 cm
Volume of
a SphereIf a sphere has a volume of V cubic units and a radius of r units, then V = 4 −
3 πr3.
Example
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Chapter 12 39 Glencoe Geometry
Skills PracticeSurface Areas and Volumes of Spheres
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
1.
7 in.
2.
32 m
3. hemisphere: radius of great circle = 8 yd
4. sphere: area of great circle ≈ 28.6 in2
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
5.
16.2 cm
6.
94.8 ft
7. hemisphere: diameter = 48 yd
8. sphere: circumference of a great circle ≈ 26 m
9. sphere: diameter = 10 in.
12-6
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Chapter 12 40 Glencoe Geometry
12-6 PracticeSurface Areas and Volumes of Spheres
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
1.
6.5 cm
2.
89 ft
3. hemisphere: radius of great circle = 8.4 in.
4. sphere: area of great circle ≈ 29.8 m2
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
5.
12.32 ft
6.32 m
7. hemisphere: diameter = 18 mm
8. sphere: circumference ≈ 36 yd
9. sphere: radius = 12.4 in.
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Chapter 12 41 Glencoe Geometry
Word Problem PracticeSurface Areas and Volumes of Spheres
1. ORANGES Mandy cuts a spherical orange in half along a great circle. If the radius of the orange is 2 inches, what is the area of the cross section that Mandy cut? Round your answer to the nearest hundredth.
2. BILLIARDS A billiard ball set consists of 16 spheres, each 2 1
−4
inches in diameter. What is the total volume of a complete set of billiard balls? Round your answer to the nearest thousandth of a cubic inch.
3. MOONS OF SATURN The planet Saturn has several moons. These can be modeled accurately by spheres. Saturn’s largest moon Titan has a radius of about 2575 kilometers. What is the approximate surface area of Titan? Round your answer to the nearest tenth.
4. THE ATMOSPHERE About 99% of Earth’s atmosphere is contained in a 31-kilometer thick layer that enwraps the planet. The Earth itself is almost a sphere with radius 6378 kilometers. What is the ratio of the volume of the atmosphere to the volume of Earth? Round your answer to the nearest thousandth.
5. CUBES Marcus builds a sphere inside ofa cube. The sphere fits snugly inside the cube so that the sphere touches the cube at one point on each side. The side length of the cube is 2 inches.
a. What is the surface area of the cube?
b. What is the surface area of the sphere? Round your answers to the nearest hundredth.
c. What is the ratio of the surface area of the cube to the surface area of the sphere? Round your answer to the nearest hundredth.
12-6
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Chapter 12 42 Glencoe Geometry
Enrichment
Spheres and DensityThe density of a metal is a ratio of its mass to its volume. For example, the mass of aluminum is 2.7 grams per cubic centimeter. Here is a list of several metals and their densities.
Aluminum 2.7 g/cm3 Copper 8.96 g/cm3
Gold 19.32 g/cm3 Iron 7.874 g/cm3
Lead 11.35 g/cm3 Platinum 21.45 g/cm3
Silver 10.50 g/cm3
To calculate the mass of a piece of metal, multiply volume by density.
Find the mass of a silver ball that is 0.8 cm in diameter.
M = D · V
= 10.5 · 4 − 3 π(0.4)3
≈ 10.5(0.27)≈ 2.81
The mass is about 2.81 grams.
ExercisesFind the mass of each metal ball described. Assume the balls are spherical. Round your answers to the nearest tenth.
1. a copper ball 1.2 cm in diameter
2. a gold ball 0.6 cm in diameter
3. an aluminum ball with radius 3 cm
4. a platinum ball with radius 0.7 cm
Solve. Assume the balls are spherical. Round your answers to the nearest tenth.
5. A lead ball weighs 326 grams. Find the radius of the ball to the nearest tenth of a centimeter.
6. An iron ball weighs 804 grams. Find the diameter of the ball to the nearest tenth of a centimeter.
7. A silver ball and a copper ball each have a diameter of 3.5 centimeters. Which weighs more? How much more?
8. An aluminum ball and a lead ball each have a radius of 1.2 centimeters. Which weighs more? How much more?
12-6
Example
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Chapter 12 43 Glencoe Geometry
12-7 Study Guide and InterventionSpherical Geometry
Geometry On A Sphere Up to now, we have been studying Euclidean geometry, where a plane is a flat surface made up of points that extends infinitely in all directions. In spherical geometry, a plane is the surface of a sphere.
Name each of the following on sphere K.
a. two lines containing the point F
� �� EG and � ��� BH are lines on sphere K that contain the point F
b. a line segment containing the point J
−−
ID is a segment on sphere K that contains the point J
c. a triangle
�AHI is a triangle on sphere K
ExercisesName two lines containing point Z, a segment containing point R, and a triangle in each of the following spheres.
1.
F
2.
M
Determine whether figure u on each of the spheres shown is a line in spherical geometry.
3. 4.
5. GEOGRAPHY Lines of latitude run horizontally across the surface of Earth. Are there any lines of latitude that are great circles? Explain.
Example
K
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Chapter 12 44 Glencoe Geometry
Comparing Euclidean and Spherical Geometries Some postulates and properties of Euclidean geometry are true in spherical geometry. Others are not true or are true only under certain circumstances.
Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.
Given any line, there are an infinite number of parallel lines.
On the sphere to the right, if we are given line m we see that it goes through the poles of the sphere. If we try to make any other line on the sphere, it would intersect line m at exactly 2 points. This property is not true in spherical geometry.A corresponding statement in spherical geometry would be: “Given any line, there are no parallel lines.”
ExercisesTell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.
1. If two nonidentical lines intersect at a point, they do not intersect again.
2. Given a line and a point on the line, there is only one perpendicular line going through that point.
3. Given two parallel lines and a transversal, alternate interior angles are congruent.
4. If two lines are perpendicular to a third line, they are parallel.
5. Three noncollinear points determine a triangle.
6. A largest angle of a triangle is opposite the largest side.
12-7 Study Guide and Intervention (continued)
Spherical Geometry
Example
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Chapter 12 45 Glencoe Geometry
Name two lines containing point K, a segment containing point T, and a triangle in each of the following spheres.
1.
C
2.
L
Determine whether figure u on each of the spheres shown is a line in spherical geometry.
3. 4.
basketball
Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.
5. If two lines form vertical angles, then the angles are equal in measure.
6. If two lines meet a third line at the same angle, those lines are parallel.
7. Two lines meet at two 90° angles or they meet at angles whose sum is 180°.
8. Three non-parallel lines divide the plane into 7 separate parts.
Skills PracticeSpherical Geometry
12-7
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Chapter 12 46 Glencoe Geometry
Name two lines containing point K, a segment containing point T, and a triangle in each of the following spheres.
1.
L
2.
M
Determine whether figure u on each of the spheres shown is a line in spherical geometry.
3.
tennis ball
4.
Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. 5. A triangle can have at most one obtuse angle.
6. The sum of the angles of a triangle is 180°.
7. Given a line and a point not on the line, there is exactly one line that goes through the point and is perpendicular to the line.
8. All equilateral triangles are similar.
9. AIRPLANES When flying an airplane from New York to Seattle, what is the shortest route: flying directly west, or flying north across Canada? Explain.
12-7 PracticeSpherical Geometry
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Chapter 12 47 Glencoe Geometry
12-7 Word Problem PracticeSpherical Geometry
1. PAINTING Consider painting quadrilateral ABCD on the beach ball with radius 1 ft. What is the surface area you would need to paint?
2. EARTH The Equator and the Prime Meridian are perpendicular great circles that divide Earth into North, South and East, West hemispheres. If Earth has a surface area of 197,000,000 square miles, what is the surface area of the North-East section of Earth?
Source: NASA
3. OCEAN If the oceans cover 70% of Earth’s surface, what is the surface area of the oceans?
Source: NASA
4. GEOMETRY Three nonidentical lines on the circle divide it into either 6 sections or 8 triangles. What condition is needed so that the three lines form 6 sections?
5. GEOGRAPHY Latitude and longitude lines are imaginary lines on Earth. The lines of latitude are horizontal concentric circles that help to define the distance a place is from the equator. Lines of latitude are measured in degrees. The equator is 0°. The north pole is 90° north latitude. The lines of longitude are great circles that help to define the distance a place is from the Prime Meridan, which is located in England and considered the longitude of 0°.
a. The mean radius of Earth is 3963 miles. Atlanta, Georgia, has coordinates (33°N, 84°W) and Cincinnati, Ohio, has coordinates (39°N, 84°W). Estimate the distance between Atlanta and Cincinnati to the nearest tenth.
b. Seattle, Washington, has coordinates (47°N, 122°W) and Portland, Oregon,has coordinates (45°N, 122°W). Estimate the distance between Portland and Seattle to the nearest tenth.
A
C
D B
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Chapter 12 48 Glencoe Geometry
12-7
ProjectionsWhen making maps of Earth, cartographers must show a sphere on a plane. To do this they have to use projections, a method of converting a sphere into a plane. But these projections have their limitations.
The map on the right is a Mercator projection of Earth. On this map Greenland appears to be the same size as Africa. But Greenland has a land area of 2,166,086 square kilometers and Africa has a land area of 30,365,700 square kilometers.
The map on the right is a Lambert projection. When a pilot draws a straight line between two points on this map the line shows true bearing, or relative direction to the North Pole. However, the bottom area of this map distorts distances.
1. When would it be useful to use a Mercator projection of Earth?
2. Does each square on the Mercator projection have the same surface area? Explain.
3. Does each square on the Lambert projection have the same surface area? Explain.
4. The Mercator projection uses a cylinder to map Earth, while the Lambert projection uses a cone to map Earth. What other shapes do you think could be used to map Earth?
EnrichmentSpherical Geometry
60˚ N
180˚ W 180˚ E90˚ W 0˚ 90˚ E
40˚ N20˚ N0˚
20˚ S40˚ S60˚ S
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Chapter 12 49 Glencoe Geometry
Identify Congruent or Similar Solids Similar solids have exactly the same shape but not necessarily the same size. Two solids are similar if they are the same shape and the ratios of their corresponding linear measures are equal. All spheres are similar and all cubes are similar. Congruent solids have exactly the same shape and the same size. Congruent solids are similar solids with a scale factor of 1:1. Congruent solids have the following characteristics:
• Corresponding angles are congruent• Corresponding edges are congruent• Corresponding faces are congruent• Volumes are equal
Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.
ratio of width: 3 − 6 = 1 −
2 ratio of length: 4 −
8 = 1 −
2
ratio of hypotenuse: 5 − 10
= 1 − 2 ratio of height: 4 −
8 = 1 −
2
The ratios of the corresponding sides are equal, so the triangular prisms are similar. The scale factor is 1:2. Since the scale factor is not 1:1, the solids are not congruent.
ExercisesDetermine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.
1.
2 cm
1 cm
10 cm
5 cm
2.
4.2 in. 12.3 in.
4.2 in.
12.3 in.
3.
4 in.
8 in.
4.
2 m
2 m
4 m1 m
1 m3 m
Study Guide and InterventionCongruent and Similar Solids
12-8
5 in.
4 in.
4 in.3 in.
10 in.
8 in.
8 in.
6 in.Example
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Chapter 12 50 Glencoe Geometry
Properties of Congruent or Similar Solids When pairs of solids are congruent or similar, certain properties are known.
If two similar solids have a scale factor of a:b then,• the ratio of their surface areas is a2:b2.• the ratio of their volumes is a3:b3.
Two spheres have radii of 2 feet and 6 feet. What is the ratio of the volume of the small sphere to the volume of the large sphere?
First, find the scale factor.
radius of the small sphere
−− radius of the large sphere
= 2 − 6 or 1 −
3
The scale factor is 1 − 3 .
a3 −
b3 = (1)3
− (3)3
or 1 − 27
So, the ratio of the volumes is 1:27.
Exercises 1. Two cubes have side lengths of 3 inches and 8 inches. What is the ratio of the surface
area of the small cube to the surface area of the large cube?
2. Two similar cones have heights of 3 feet and 12 feet. What is the ratio of the volume of the small cone to the volume of the large cone?
3. Two similar triangular prisms have volumes of 27 square meters and 64 square meters. What is the ratio of the surface area of the small prism to the surface area of the large prism?
4. COMPUTERS A small rectangular laptop has a width of 10 inches and an area of 80 square inches. A larger and similar laptop has a width of 15 inches. What is the length of the larger laptop?
5. CONSTRUCTION A building company uses two similar sizes of pipes. The smaller size has a radius of 1 inch and length of 8 inches. The larger size has a radius of 2.5 inches What is the volume of the larger pipes?
Study Guide and Intervention (continued)
Congruent and Similar Solids
Example
12-8
2 ft
6 ft
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Chapter 12 51 Glencoe Geometry
Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.
1.
2 cm
4 cm
3 cm
6 cm
12 cm
9 cm 2.
6 cm
9 cm
8 cm
12 cm
3.
5 m 10 m
4.
3 ft
1 ft1 ft 9 ft
3 ft3 ft
5. Two similar pyramids have heights of 4 inches and 7 inches What is the ratio of the volume of the small pyramid to the volume of the large pyramid?
6. Two similar cylinders have surface areas of 40π square feet and 90π square feet. What is the ratio of the height of the large cylinder to the height of the small cylinder?
7. COOKING Two stockpots are similar cylinders. The smaller stockpot has a height of 10 inches and a radius of 2.5 inches. The larger stockpot has a height of 16 inches. What is the volume of the larger stockpot? Round to the nearest tenth.
Skills PracticeCongruent and Similar Solids
12-8
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Chapter 12 52 Glencoe Geometry
Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.
1.
6 cm
8 cm
18 cm
24 cm
2.
5 cm
12 cm
10 cm
24 cm
3. 1 m 1 m5 m 5 m
3 m 3 m
4 m4 m
4.
5 cm
5 cm
2 cm
10 cm
10 cm
1.5 cm
5. Two cubes have surface areas of 72 square feet and 98 square feet. What is the ratio of the volume of the small cube to the volume of the large cube?
6. Two similar ice cream cones are made of a half sphere on top and a cone on bottom. They have radii of 1 inch and 1.75 inches respectively. What is the ratio of the volume of the small ice cream cone to the volume of the large ice cream cone? Round to the nearest tenth.
7. ARCITHECTURE Architects make scale models of buildings to present their ideas to clients. If an architect wants to make a 1:50 scale model of a 4000 square foot house, how many square feet will the model have?
12-8 PracticeCongruent and Similar Solids
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Chapter 12 53 Glencoe Geometry
12-8 Word Problem PracticeCongruent and Similar Solids
1. COOKING A cylindrical pot is 4.5 inchestall and has a radius of 4 inches. How tall would a similar pot be if its radius is 6 inches?
2. MANUFACTURING Boxes, Inc. wants to make the two boxes below. How long does the second box need to be so that they are similar?
24 cm
15 cm 25 cm
25 cm15 cm
3. FARMING A farmer has two similar cylindrical grain silos. The smaller silo is 25 feet tall and the larger silo is 40 feet tall. If the smaller silo can hold 1500 cubic feet of grain, how much can the larger silo hold?
4. PLANETS Earth has a surface area of about 196,937,500 square miles. Mars has a surface area of about 89,500,000 square miles. What is the ratio of the radius of Earth to the radius of Mars? Round to the nearest tenth.
Source: NASA
5. BASEBALL Major League Baseball or MLB, rules state that baseballs must have a circumference of 9 inches. The National Softball Association, or NSA, rules state that softballs must have a circumference not exceeding 12 inches.
Source: MLB, NSA
a. Find the ratio of the circumference of MLB baseballs to the circumference of NSA softballs.
b. Find the ratio of the volume of MLB baseballs to the volume of NSA softballs. Round to the nearest tenth.
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Chapter 12 54 Glencoe Geometry
Enrichment
Doubling SizesConsider what happens to surface area when the sides of a figure are doubled.
The sides of the large cube are twice the size of the sides of the small cube.
1. How long are the edges of the large cube?
2. What is the surface area of the small cube?
3. What is the surface area of the large cube?
4. The surface area of the large cube is how many times greater than that of the small cube?
The radius of the large sphere at the right is twice the radius of the small sphere.
5. What is the surface area of the small sphere?
6. What is the surface area of the large sphere?
7. The surface area of the large sphere is how many times greater than the surface area of the small sphere?
8. It appears that if the dimensions of a solid are doubled, the surface area is multiplied by .
Now consider how doubling the dimensions affects the volume of a cube.
The sides of the large cube are twice the size of the sides of the small cube.
9. How long are the edges of the large cube?
10. What is the volume of the small cube?
11. What is the volume of the large cube?
12. The volume of the large cube is how many times greater than that of the small cube?
The large sphere at the right has twice the radius of the small sphere.
13. What is the volume of the small sphere?
14. What is the volume of the large sphere?
15. The volume of the large sphere is how many times greater than the volume of the small sphere?
16. It appears that if the dimensions of a solid are doubled, the volume is multiplied by .
5 in.
3 m
5 in.
3 m
12-8
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Chapter 12 55 Glencoe Geometry
12 Student Recording SheetUse this recording sheet with pages 894– 895 of the Student Edition.
Read each question. Then fill in the correct answer.
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
Multiple Choice
Extended Response
Record your answers for Question 13 on the back of this paper.
9.
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing each number or symbol in a box. Then fill in the corresponding circle for that number or symbol.
7. ————————
8. ————————
9. ———————— (grid in)
10. ————————
11. ———————— (grid in)
12. ————————
Short Response/Gridded Response
9
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3
2
1
0
9
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. . . . .
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Chapter 12 56 Glencoe Geometry
12
General Scoring Guidelines
• If a student gives only a correct numerical answer to a problem but does not show how he or she arrived at the answer, the student will be awarded only 1 credit. All extended response questions require the student to show work.
• A fully correct answer for a multiple-part question requires correct responses for all parts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.
• Students who use trial and error to solve a problem must show their method. Merely showing that the answer checks or is correct is not considered a complete response for full credit.
Exercise 13 Rubric
Score Specific Criteria
4 A correct solution that is supported by well-developed, accurate explanations. The scale factor of the prisms, and the correct volumes of the prisms are provided. The student displays an understanding of how volume changes based upon changing dimensions by correctly answering parts c and d.
3 A generally correct solution, but may contain minor flaws in reasoning or computation.
2 A partially correct interpretation and/or solution to the problem.
1 A correct solution with no evidence or explanation.
0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.
Rubric for Scoring Extended-Response
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Chapter 12 57 Glencoe Geometry
12
12
1. Given the corner view of a figure, draw the left view.
2. A cylinder has a lateral area of 120π square meters, and a height of 7 meters. Find the radius. Round to the nearest tenth.
3. Find the lateral area of the hexagonal prism.
4. Find the surface area of a rectangular prism with a length and width of 6 centimeters and a height of 12 centimeters.
5. MULTIPLE CHOICE Find the surface area of the prism to the nearest hundredth.
A 30.50 B 54.00
C 49.45 D 52.44
1.
2.
3.
4.
5.
1. Find the surface area of the solid figureat the right to the nearest tenth.
For Questions 2 and 3, use a right circular cone with a radius of 5 feet and a slant height of 12 feet. Round to the nearest tenth.
2. Find the lateral area.
3. Find the surface area.
4. A rectangular prism has a length of 16 feet, a width of 9 feet, and a height of 8 feet. Find the volume of the prism.
5. A cylinder has a diameter of 20 inches and a height of 9 inches. Find the volume of the cylinder, round to the nearest tenth.
Chapter 12 Quiz 1(Lessons 12-1 and 12-2)
Chapter 12 Quiz 2(Lessons 12-3 and 12-4)
3.5
35
210
6 in.
9 in.9 in.
1.
2.
3.
4.
5.
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Chapter 12 58 Glencoe Geometry
12 Chapter 12 Quiz 3(Lessons 12-5 and 12-6)
1. A pyramid has a height of 18 centimeters and a base with an area of 26 square centimeters. Find the volume.
2. Find the volume of the cone. Round to the nearest tenth.
3. A hemisphere has a base with an area that is 25π square centimeters. Find the volume of the hemisphere. Round to the nearest tenth.
4. A sphere has a great circle with a circumference of 8π meters. What is the surface area of the sphere?
5. MULTIPLE CHOICE A sphere has a radius that is 15.6 inches long. Find the volume of the sphere. Round to the nearest tenth.
A 1019.4 in3 C 15,902 in3
B 7951.2 in3 D 47,702.2 in3
1.
2.
3.
4.
5.
1. Name two lines containing point Z, a segment containing point R, and a triangle in the sphere F.
2. Do all lines have an infinite number of points in spherical geometry? If not, explain your reasoning.
3. Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.
4. Two similar prisms have heights of 12 feet and 20 feet. What is the ratio of the volume of the small prism to the volume of the large prism?
5. Two cubes have surface areas of 81 square inches and 144 square inches. What is the ratio of the volume of the small cube to the volume of the large cube?
1.
2.
3.
4.
5.
Chapter 12 Quiz 4(Lessons 12-7 and 12-8)
12
16 cm
17 cm
F
1 m 0.5 m
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Chapter 12 59 Glencoe Geometry
12 Chapter 12 Mid-Chapter Test(Lessons 12-1 through 12-4)
1.
2.
3.
4.
5.
1. A cylinder is standing on one of its bases. It is sliced by a plane horizontally. What is the shape of the cross section?
A triangle C circle B square D rectangle
2. Choose the correct formula for the surface area of a cone. F S = Ph + 2B H S = 1 −
2 P� + B
G S = πr� + πr2 J S = πr� + 2πr
3. The surface area of a prism is 120 square centimeters and the area of each base is 32 square centimeters. Find the lateral area of the prism.A 184 cm2 B 152 cm2 C 86 cm2 D 56 cm2
For Questions 4 and 5, refer to the solid figure. Round to the nearest tenth. 4. Find the lateral area.
F 9289.1 ft2 H 10,965.4 ft2
G 9434.2 ft2 J 12,641.8 ft2
5. Find the surface area.A 9289.1 ft2 B 9434.2 ft2 C 10,965.4 ft2 D 12,641.8 ft2
6.
7.
8.
9.
10.
Part II
6. Draw the top view of this orthogonal drawing.
For Questions 7 and 8, refer to the regular hexagonal prism.
7. Sketch the cross section from a vertical slice of the figure.
8. Find the surface area. Round to the nearest tenth.
9. A barrel in the shape of a right cylinder has a diameter of 18 inches and a height of 42 inches. Find the surface area of the barrel.
10. Find the lateral area of the solid. Round to the nearest tenth.
Part I Write the letter for the correct answer in the blank at the right of each question.
4 in.
6 in.
2 6
46.2 ft
64 ft
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Chapter 12 60 Glencoe Geometry
12 Chapter 12 Vocabulary Test
altitude
axis
base edges
composite solid
congruent solid
cross section
Euclidean geometry
great circle
isometric view
lateral area
lateral edge
lateral face
non-Euclidean geometry
oblique cone
oblique cylinder
oblique prism
regular pyramidright cone
right cylinder
right prism
similar solids
slant height
spherical geometry
topographical map
Choose from the terms above to complete each sentence.
1. The height of each lateral face of a regular pyramid is called a(n) .
2. have the same shape but not the same size.
3. If the axis of a cylindrical is also the altitude, then the cylinder is called a(n) .
4. is the measure of the amount of space that a figure encloses.
Choose the correct term to complete each sentence.
5. The segment whose endpoints are the centers of the circular bases of a cylinder is the (axis, hemisphere)
6. Two solids that have the same shape and the same size are called (congruent solids, composite solids).
State whether each sentence is true or false. If false, replace the underlined word or phrase to make a true sentence.
7. A polyhedron that has all but one face intersecting at one point is a prism.
8. A cross section is the intersection of a plane and a solid figure.
9. A hexagonal prism has a six lateral faces.
10. A cone with an axis that is not an altitude is a right cone.
Define each term in your own words.
11. lateral area
12. right prism
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
?
?
?
?
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Chapter 12 61 Glencoe Geometry
12 Chapter 12 Test, Form 1
Write the letter for the correct answer in the blank at the right of each question.
1. Which of these is part of an orthographic drawing? A a perspective view C a two-dimensional top view B a corner view D a three-dimensional view
For Questions 2–4, refer to the figure.
2. Identify this solid figure. F square pyramid H triangular pyramid G square prism J triangular prism
3. Name the base. A �ABE B �ABCD C �CDE D E
4. The shape of a vertical cross section of a cone is a . F circle G triangle H square J trapezoid
5. Find the surface area of the cube. A 9 in2 C 36 in2
B 27 in2 D 54 in2
For Questions 6 and 7, refer to the figure.
6. Find the lateral area. Round to the nearest tenth. F 75.4 ft2 H 50.3 ft2
G 62.8 ft2 J 25.1 ft2
7. Find the surface area. Round to the nearest tenth. A 75.4 ft2 B 62.8 ft2 C 50.3 ft2 D 25.1 ft2
For Questions 8 and 9, refer to the figure.
8. Find the lateral area. F 108 cm2 H 162 cm2
G 144 cm2 J 324 cm2
9. Find the surface area. A 108 cm2 B 144 cm2 C 162 cm2 D 324 cm2
10. Find the surface area to the nearest tenth. F 546.6 units2 H 1017.9 units2
G 989.6 units2 J 1046.2 units2
11. The radius of a cone is 17 inches long and the slant height is 20 inches. Find the surface area to the nearest tenth.
A 18,158.4 in2 B 1976.1 in2 C 1068.1 in2 D 340 in2
E
AB
D
C
3 in.3 in.
3 in.
2 ft
4 ft
50
5
6
1.
2.
3.
4.
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7.
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?
6 cm6 cm
9 cm
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Chapter 12 62 Glencoe Geometry
12 Chapter 12 Test, Form 1 (continued)
12. The area of the base of a prism is 96 square centimeters and the height is 9 centimeters. Find the volume of the prism.
F 288 cm3 G 864 cm3 H 932 cm3 J 7776 cm3
13. The volume of a cylinder is 62.8 cubic meters and the radius is 2 meters. Find the height of the cylinder. Round to the nearest meter.
A 20 m B 10 m C 8 m D 5 m
14. A pyramid has a height of 10 inches and a base with an area of 21 square inches. Find the volume of the pyramid.
F 210 in3 G 105 in3 H 70 in3 J 35 in3
15. Find the volume of the oblique cone. Round to the nearest tenth.
A 1206.4 in3 C 301.6 in3
B 402.1 in3 D 100.5 in3
16. The diameter of a sphere is 42 centimeters. Find the surface area to the nearest tenth.
F 5541.8 cm2 G 2770.9 cm2 H 2167.1 cm2 J 527.8 cm2
17. A sphere has a volume that is 36π cubic meters. Find the radius of the sphere.
A 2 m B 3 m C 6 m D 12 m
18. Which of the following postulates or properties of spherical geometry are true?
F A line has an infinite number of lines parallel to it. G No two lines are parallel. H Alternate interior angles formed by two parallel lines and a
transversal are equal in measure. J Four noncollinear points form two parallel lines.
19. Which of the following describes the two spheres? A congruent C both congruent and similar B similar D neither congruent nor similar
20. The ratio of the side lengths of two cubes is 3:7. Find the ratio of their volumes.
F 3:7 G 9:21 H 9:49 J 27:343
Bonus Find the amount of glass needed to cover the sides of the greenhouse shown. The bottom, front, and back are not glass.
12.
13.
14.
15.
16.
17.
18.
19.
20.
15 ft
30 ft
9 ft
9 ft
8 ft8 ft
B:
6 in.
10 in.
9 ft 6 ft
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Chapter 12 63 Glencoe Geometry
12
Write the letter for the correct answer in the blank at the right of each question.
1. What do the dark segments represent in an orthographic drawing? A changes in color C designs on the surface B where paper should be folded D different heights in the surface
For Questions 2 and 3, refer to the figure. 2. Identify the figure. F pyramid H cone G prism J cylinder
3. Identify the shape of a horizontal cross section of the figure. A triangle B ellipse C rectangle D circle
4. The lateral area of a cube is 36 square inches. How long is each edge? F √ � 6 in. G 3 in. H 6 in. J 9 in.
5. Find the surface area of the outside of the open box. A 1920 in2 C 752 in2
B 998 in2 D 400 in2
For Questions 6 and 7, use a right cylinder with a radius of 3 inches and a height of 17 inches. Round to the nearest tenth. 6. Find the lateral area. F 320.4 in2 G 348.7 in2 H 377.0 in2 J 537.2 in2
7. Find the surface area. A 320.4 in2 B 348.7 in2 C 377.0 in2 D 537.2 in2
For Questions 8 and 9, refer to the figure. 8. Find the lateral area. F 144 cm2 H 196 cm2
G 144 + 24 √ � 3 cm2 J 288 cm2
9. Find the surface area. A 144 cm2 B 144 + 24 √ � 3 cm2 C 196 cm2 D 288 cm2
For Questions 10 and 11, refer to the figure. Round to the nearest tenth.
10. Find the lateral area. F 44.0 in2 G 75.4 in2 H 88.0 in2 J 100.5 in2
11. Find the surface area. A 44.0 in2 B 75.4 in2 C 88.0 in2 D 100.5 in2
Chapter 12 Test, Form 2A
Y
X
12 cm
4 cm
12 in.2 in.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
20 in.12 in.
8 in.
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Chapter 12 64 Glencoe Geometry
12
12. The surface area of a cube is 96 square feet. Find the volume of the cube.
F 4 ft3 G 16 ft3 H 64 ft3 J 256 ft3
13. A cylinder whose height is 5 meters has a volume of 320π cubic meters. Find the radius of the cylinder.
A 8 m B 12.8 m C 64 m D 201 m
14. A square pyramid has a height that is 8 centimeters long and a base with sides that are each 9 centimeters long. Find the volume of the pyramid.
F 648 cm3 G 324 cm3 H 216 cm3 J 162 cm3
15. Find the volume to the nearest tenth. A 3619.1 m3 C 14,476.5 m3
B 4825.5 m3 D 43,429.4 m3
16. Find the surface area to the nearest tenth. F 4536.5 m2 H 477.5 m2
G 2268.2 m2 J 238.8 m2
17. A sphere has a volume of 972π cubic inches. Find the radius of the sphere.
A 2 in. B 3 in. C 6 in. D 9 in.
18. The shortest distance between any two points in spherical geometry is
F a straight line. G any circle. H a great circle. J a line through the sphere.
19. Two square pyramids are similar. The sides of the bases are 4 inches and 12 inches. The height of the smaller pyramid is 6 inches. Find the height of the larger pyramid.
A 24 in. B 18 in. C 16 in. D 14 in.
20. The ratio of the radii of two similar cylinders is 3:5. The volume of the smaller cylinder is 54π cubic centimeters. Find the volume of the larger cylinder.
F 90π cm3 G 150π cm3 H 250π cm3 J 540π cm3
Bonus Find the surface area of the figure to the nearest tenth.
Chapter 12 Test, Form 2A (continued)
4 ft
12 ft
1 ft
60°
24 m
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
19 m
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Chapter 12 65 Glencoe Geometry
12
Write the letter for the correct answer in the blank at the right of each question.
1. Given the corner view of a figure, which is the top view? A B C D
For Questions 2 and 3, refer to the figure. 2. Identify the figure. F pyramid H cone G prism J cylinder
3. Identify the shape of a vertical cross section of the figure. A rectangle B circle C triangle D parabola
4. Find the lateral area of an equilateral triangular prism if the area of each lateral face is 10 square centimeters.
F 10 √ � 3 cm2 G 30 cm2 H 50 cm2 J 100 cm2
5. The surface area of a rectangular prism is 190 square inches, the length is 10 inches, and the width 3 inches. Find the height.
A 30 in. B 20 in. C 10 in. D 5 in.
For Questions 6–9, use a right cylinder with a radius of 5 centimeters and a height of 22 centimeters. Round to the nearest tenth. 6. Find the lateral area. F 848.2 cm2 G 769.7 cm2 H 691.2 cm2 J 345.6 cm2
7. Find the surface area. A 848.2 cm2 B 769.7 cm2 C 691.2 cm2 D 345.6 cm2
8. Find the volume. F 345.6 cm3 G 691.2 cm3 H 1727.9 cm3 J 2290.2 cm3
For Questions 9–11, refer to the figure. Round to the nearest tenth. 9. Find the lateral area. A 75.4 cm2 C 131.9 cm2
B 103.7 cm2 D 150.8 cm2
10. Find the surface area. G 75.4 cm2 G 103.7 cm2 H 131.9 cm2 J 150.8 cm2
11. Find the volume. A 50.3 cm3 B 69.9 cm3 C 209.7 cm3 J 226.2 cm3
Chapter 12 Test, Form 2B
N
M
3 cm8 cm
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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Chapter 12 66 Glencoe Geometry
12 Chapter 12 Test, Form 2B (continued)
12. The lateral area of a cube is 324 square centimeters. Find the volume of the cube.
F 9 cm3 G 81 cm3 H 729 cm3 J 972 cm3
13. Find the volume of the solid. Round to the nearest tenth.A 31.4 in3 C 125.7 in3
B 41.9 in3 D 502.7 in3
14. A right triangular pyramid has a 12-meter height and a base with legs that are 3 meters and 4 meters long. Find the volume of the triangular pyramid.
F 144 m3 G 72 m3 H 48 m3 J 24 m3
15. Find the volume of the cone. Round to the nearest tenth.A 41,224.0 m3 C 10,306.0 m3
B 20,612.0 m3 D 763.4 m3
16. The surface area of a sphere is 64π square centimeters. Find the radius. F 16 cm G 8 cm H 4 cm J 2 cm
17. A sphere has a 48-centimeter diameter. Find the volume of the sphere. Round to the nearest tenth.
A 463,246.7 cm3 B 57,905.8 cm3 C 28,952.9 cm3 D 7238.2 cm3
18. In spherical geometry, two lines must meet at least how many times?F zero times H two times G one time J an infinite number of times
19. Find the scale factor between the two similar cones.
A 3:8 C 1:2 B 1:3 D 1:4
20. The ratio of the heights of two similar solids is 6:11. Find the ratio of their surface areas.
F 6:11 G 36:121 H 216:1331 J 24:44
Bonus Find the surface area of the frustum of a square pyramid.
10 in.
4 in.
45°
27 m
6 ft
16 ft8 ft
3 ft
3 ft
2 ft
4 ft
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
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Chapter 12 67 Glencoe Geometry
12 Chapter 12 Test, Form 2C
1. Given the corner view of a figure, sketch the front view.
2. Name the faces of the solid.
3. Sketch the shape of a horizonal cross section of the solid.
4. Find the lateral area of a triangular prism with a height of 8 centimeters, and with bases having sides that measure 4 centimeters, 5 centimeters, and 6 centimeters.
5. Find the surface area of the solid.
6. Find the lateral area of a right cylinder with a diameter of 8.6 yards and a height of 19.4 yards. Round to the nearest tenth.
7. The surface area of a cylinder is 180π square inches and the height is 9 inches. Find the radius.
For Questions 8 and 9, use a regular hexagonal pyramid with base edges of 10 inches and a slant height of 9 inches.
8. Find the lateral area.
9. Find the surface area.
For Questions 10 and 11, use a right circular cone with a radius of 4 feet and a height of 3 feet. Round to the nearest tenth.
10. Find the lateral area.
11. Find the surface area.
1 in.
1 in.
1 in.
P S
RQ
T
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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Chapter 12 68 Glencoe Geometry
12
12. The volume of a rectangular prism is 120 cubic feet and the area of the base is 60 square feet. Find the length of a lateral edge of the prism.
13. A cylinder has a 12-foot radius and a 17-foot height. Find the volume of the cylinder. Round to the nearest tenth.
14. A regular hexagonal pyramid has a height that is 15 feet and a base 6 feet on each side. Find the volume of the pyramid. Round to the nearest tenth.
15. Find the volume of the oblique cone. Round to the nearest tenth.
16. Find the surface area of this hemisphere to the nearest tenth.
17. A sphere has a diameter of 7.36 inches long. Find the volume of the sphere. Round to the nearest tenth.
18. What best describes a line in spherical geometry?
19. Determine whether these two cylinders are congruent, similar, or neither.
20. The ratio of the heights of two similar prisms is 2:7. The surface area of the smaller prism is 50 square meters. Find the surface area of the larger prism.
Bonus The length of each side of a cube is 6 inches long. Find the surface area of a sphere inscribed in the cube. Round to the nearest tenth.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 12 Test, Form 2C (continued)
11 in.
30°
8 ft 5 ft
4 in.
10 in. 9 in.
3 in.
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Chapter 12 69 Glencoe Geometry
12
1. Given the corner view of a figure, sketch the back view.
2. Name the edges of the solid.
3. Sketch the shape of a horizonal cross section of the solid.
4. Find the lateral area of a regular pentagonal prism if the perimeter of the base is 50 inches and the height is 15 inches.
5. Find the surface area of the prism.
6. A right cylinder has a diameter of 23.6 meters and a height of 11.4 meters. Find the lateral area of the cylinder. Round to the nearest tenth.
7. The surface area of a right cylinder is 252π square feet and the height is 11 feet. Find the radius of the cylinder.
For Questions 8 and 9, use a regular octagonal pyramid with base edges 9 feet long, slant height 15 feet, and a base with an apothem of 10.86 feet. 8. Find the lateral area.
9. Find the surface area to the nearest tenth.
For Questions 10 and 11, use a cone with a radius of 5 centimeters and a height of 12 centimeters. Round to the nearest tenth. 10. Find the lateral area.
11. Find the surface area.
Chapter 12 Test, Form 2D
9 ft5 ft
6 ft
G
I
H
J
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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Chapter 12 70 Glencoe Geometry
12
12. An aquarium is 18 inches long, 8 inches wide, and 14 inches high. The water in it is 4 inches deep. Find the volume of the water.
13. Find the volume of the cylinder. Round to the nearest tenth.
14. A square pyramid has a height that is 51 inches and a base with sides that are each 11 inches long. Find the volume of the pyramid.
15. Find the volume of the oblique cone. Round to the nearest tenth.
16. Find the surface area of the hemisphere. Round to the nearest tenth.
17. A sphere has a radius that is 2.94 centimeters long. Find the volume of the sphere. Round to the nearest tenth.
18. Why do parallel lines not exist in spherical geometry?
19. Determine whether these cubes are congruent, similar, or neither.
20. The ratio of the heights of two similar pyramids is 2:5 and the volume of the smaller pyramid is 100 cubic feet. Find the volume of the larger pyramid.
Bonus The length of each side of a cube is 8 inches long. Find the surface area of a sphere inscribed in the cube. Round to the nearest tenth.
13 in.
Chapter 12 Test, Form 2D (continued)
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
25 cm7 cm
4 cm
6 cm
6 cm 8 cm
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Chapter 12 71 Glencoe Geometry
12
1. Draw the back view of a figure given its orthographic drawing.
2. Describe the cross section.
3. Find the surface area of the solid. Show the exact solution.
4. Find the lateral area of a triangular prism with a right triangular base with legs that measure 2 feet and 3 feet and a height of 7 feet. Show the exact solution.
5. Find the surface area of the prism.
For Questions 6 and 7, use a right cylinder with a diameter of 96.4 feet and a height of 58.9 feet. Round to the nearest tenth.
6. Find the lateral area.
7. Find the surface area.
For Questions 8 and 9, refer to the solid. Round to the nearest tenth if necessary. 8. Find the lateral area.
9. Find the surface area to the nearest tenth.
For Questions 10 and 11, use a right circular cone with a radius of 7 inches and a height of 8 inches. Round to the nearest tenth. 10. Find the lateral area.
11. Find the surface area.
Chapter 12 Test, Form 3
top view left view front view right view 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
13 in.
12 in.
84
3
21√
3
6
10
20
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Chapter 12 72 Glencoe Geometry
12
12. Find the volume of the solid.
13. The volume of a cylinder is 96π cubic meters and the height is 6 meters. Find the length of the diameter of this cylinder.
14. Sam is filling a rectangular pan with liquid from a cylindrical can. The can is three-fourths full of water. Determine whether all of the water will fit in the pan. Explain.
15. Find the volume of the solid.
16. Write a formula for the surface area of a hemisphere in terms of π and the radius r.
17. A cone is 9 centimeters deep and 4 centimeters across the top. A single scoop of ice cream, 4 centimeters in diameter, is placed on top of the cone. If the ice cream melts into the cone, determine whether the melted ice cream will fit in the cone. Explain.
18. Is the angle sum of a triangle in spherical geometry always 180°? If not give an example.
19. Determine whether these two pyramids are congruent, similar, or neither.
20. The ratio of the volumes of two similar solids is 1:2. Find the ratio of their surface areas.
Bonus Find the surface area of the solid to the nearest square foot. Do not include the area of the base.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
8 ft
10 ft
6 ft
Chapter 12 Test, Form 3 (continued)
3 cm2 cm
8 cm
10 cm
12 cm
8 in.6 in.
7 in.3 in.2 in.
2 in. 8 in.
13 in.
5 m
5 m
8 m
4 m
5 m8 m
6.4 m8 m
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Chapter 12 73 Glencoe Geometry
12 Chapter 12 Extended-Response Test
Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.
1. Explain the difference between the lateral area and the surface area of a prism.
2. Draw an oblique cylinder and a right cylinder.
3. Write a practical application problem involving the surface area or lateral area of a solid figure studied in this chapter.
4. Give the dimensions of two cylinders in which the first has a greater volume than the second, but the second has greater surface area than the first.
5. Draw and label the dimensions of a prism and a pyramid that have the same volume.
6. Write a formula for the volume of this solid in terms of the radius r. Explain.
45°
2r
r
r
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Chapter 12 74 Glencoe Geometry
12
x
yxO
B
A F G
C
E
D
97°
49°
8.9
r
Q R
P
40°
40°
4.2
12.6
9.5
a
A
B E
C
F 1. Which method could you use to prove
−−−
BE � −−
AC if AF = BF? (Lesson 4-5)
A Show that �ABE � �BAC by SSS, then
−−−
BE � −−
AC by CPCTC. B Show that �ABE � �BAC by ASA, then
−−−
BE � −−
AC by CPCTC. C Show that �BFE � �AFC by SAS, then
−−−
BE � −−
AC by CPCTC. D Show that �ABE � �BAC by AAS, then
−−−
BE � −−
AC by CPCTC.
2. Find a. (Lesson 7-3)
F 28.5 H 12.6G 6.3 J 14
3. Find r. (Lesson 8-6) A about 34.0 C about 11.8B about 8.9 D about 6.6
4. A square has side length 18 centimeters. Find the area of the square. (Lesson 11-1)
F 36 cm2 G 40 cm2 H 81 cm2 J 324 cm2
5. What can you assume from the figure? (Lesson 10-3)
A �ABC is isosceles.B �ABC is equilateral.C DF = EGD radius of �O = x + y
6. Points D, E, and F are on a circle so that m � DEF = 210. If K is the center of the circle, what is m ∠DKF? (Lesson 10-2)
F 210 G 105 H 70 J 35
7. Which net could be folded into a triangular prism? (Lesson 12-1)
A B C D
8. Find the surface area of a square pyramid with a height of 9 centimeters and base with a side measuring 24 centimeters. (Lesson 12-3)
F 1296 cm2 G 1806 cm2 H 2016 cm2 J 8640 cm2
1.
2. F G H J
3. A B C D
4. F G H J
5.
6. F G H J
7. A B C D
8. F G H J
A B C D
Standardized Test Practice(Chapters 1–12)
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
A B C D
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Chapter 12 75 Glencoe Geometry
12
13.
14.
9. Find y to the nearest centimeter. (Lesson 8-6)
A 19 cm C 34 cmB 28 cm D 37 cm
10. A plane figure is the locus of all points in a plane equidistant from point B. What is the shape of this figure? (Lesson 10-1)
F square G cylinder H rhombus J circle
11. Find m∠C. (Lesson 10-6)
A 18º C 28ºB 25º D 60º
12. Find the area of the figure. (Lesson 11-1)
F 76 cm2 H 88 cm2
G 80 cm2 J 92 cm2
73°
32 cm
25 cmy
Z Y
X
9.
10.
11.
12. F G H J
A B C D
F G H J
A B C D
13. Quadrilateral PQSR is a rectangle. Find a. (Lesson 6-4)
14. Find x. Assume that segments that appear tangent are tangent.(Lesson 10-5)
2x + 8
5x - 7
M
N
(2a - 1)°(6a - 21)°
P Q
SR
Standardized Test Practice (continued)
6 cm
4 cm
4 cm
7 cm
3 cm2 cm
Part 2: Gridded Response
Instructions: Enter your answer by writing each digit of the answer in a column box
and then shading in the appropriate circle that corresponds to that entry.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
88° 32°
E
A
D
BC
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Chapter 12 76 Glencoe Geometry
12
5√3 cm
60° 60°
60°
15 cm
15
3018
x
50°
3 mT
S
R
15. Find the length of � SR to the nearest tenth. (Lesson 10-2)
16. Find x. (Lesson 10-7)
17. Find the area of the shaded region to the nearest tenth. (Lesson 11-3)
18. Identify the solid. (Lesson 12-1)
19. A right circular cone has a slant height of 15 inches and a radius that is 25 inches long. Find the surface area of the cone. Round to the nearest tenth. (Lesson 12-3)
20. A ball has a diameter of 26.5 centimeters. Find the surface area of the ball. Round to the nearest tenth. (Lesson 12-6)
21. Find the following measurements for a sphere with a diameter of 66 meters. Round to the nearest tenth. (Lesson 12-6)
a. surface area
b. circumference of the great circle
c. area of the great circle
d. surface area of the hemisphere
15.
16.
17.
18.
19.
20.
21a.
b.
c.
d.
Standardized Test Practice (continued)
Part 3: Short Response
Instructions: Write your answer in the space provided.
066-076_GEOCRMC12_890521.indd 76066-076_GEOCRMC12_890521.indd 76 6/1/09 11:58:42 AM6/1/09 11:58:42 AM
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swer
s
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ight
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oe/
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The
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raw
-Hill
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mp
anie
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c.
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Chapter 12 A1 Glencoe Geometry
Chapter Resources
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ME
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Cha
pte
r 12
3
G
lenc
oe G
eom
etry
Bef
ore
you
beg
in C
ha
pte
r 12
• R
ead
each
sta
tem
ent.
• D
ecid
e w
het
her
you
Agr
ee (
A)
or D
isag
ree
(D)
wit
h t
he
stat
emen
t.
• W
rite
A o
r D
in
th
e fi
rst
colu
mn
OR
if
you
are
not
su
re w
het
her
you
agr
ee o
r di
sagr
ee, w
rite
NS
(N
ot S
ure
).
Aft
er y
ou c
omp
lete
Ch
ap
ter
12
• R
erea
d ea
ch s
tate
men
t an
d co
mpl
ete
the
last
col
um
n b
y en
teri
ng
an A
or
a D
.
• D
id a
ny
of y
our
opin
ion
s ab
out
the
stat
emen
ts c
han
ge f
rom
th
e fi
rst
colu
mn
?
• F
or t
hos
e st
atem
ents
th
at y
ou m
ark
wit
h a
D, u
se a
pie
ce o
f pa
per
to w
rite
an
exam
ple
of w
hy
you
dis
agre
e.
12A
ntic
ipat
ing
Gui
deE
xte
nd
ing
Su
rface A
rea a
nd
Vo
lum
e
Step
1
ST
EP
1A
, D
, o
r N
SS
tate
men
tS
TE
P 2
A o
r D
1.
Th
e sh
ape
of a
hor
izon
tal
cros
s se
ctio
n o
f a
squ
are
pyra
mid
is
a t
rian
gle.
2.
Th
e la
tera
l ar
ea o
f a
pris
m i
s eq
ual
to
the
sum
of
the
area
s of
eac
h f
ace.
3.
Th
e ax
is o
f an
obl
iqu
e cy
lin
der
is d
iffe
ren
t th
an t
he
hei
ght
of t
he
cyli
nde
r.
4.
Th
e sl
ant
hei
ght
and
hei
ght
of a
reg
ula
r py
ram
id a
re
the
sam
e.
5.
Th
e la
tera
l ar
ea o
f a
con
e eq
ual
s th
e pr
odu
ct o
f π
, th
e ra
diu
s, a
nd
the
hei
ght
of t
he
con
e.
6.
Th
e vo
lum
e of
a r
igh
t cy
lin
der
wit
h r
adiu
s r
and
hei
ght
h i
s πr
2 h.
7.
Th
e vo
lum
e of
a p
yram
id o
r a
con
e is
fou
nd
by m
ult
iply
ing
the
area
of
the
base
by
the
hei
ght.
8.
To
fin
d th
e su
rfac
e ar
ea o
f a
sph
ere
wit
h r
adiu
s r,
m
ult
iply
πr2
by 4
.
9.
All
pos
tula
tes
and
prop
erti
es o
f E
ucl
idea
n g
eom
etry
are
tru
e in
sp
her
ical
geo
met
ry.
10.
All
sph
eres
an
d al
l cu
bes
are
sim
ilar
sol
ids.
Step
2
D D A D D A D A D A
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Lesson 12 -1
Cha
pte
r 12
5
G
lenc
oe G
eom
etry
Dra
w Is
om
etri
c V
iew
s Is
omet
ric
dot
pape
r ca
n b
e u
sed
to d
raw
iso
met
ric
view
s, o
r co
rner
vie
ws,
of
a th
ree-
dim
ensi
onal
obj
ect
on t
wo-
dim
ensi
onal
pap
er.
U
se i
som
etri
c d
ot p
aper
to
sket
ch a
tri
angu
lar
pri
sm 3
un
its
hig
h, w
ith
tw
o si
des
of
the
bas
e th
at a
re 3
un
its
lon
g an
d 4
un
its
lon
g.S
tep
1
Dra
w −
−
A
B a
t 3
un
its
and
draw
−−
A
C a
t 4
un
its.
Ste
p 2
D
raw
−−
−
A
D ,
−−
−
B
E , a
nd
−−
C
F , e
ach
at
3 u
nit
s.S
tep
3
Dra
w −
−−
B
C a
nd
�D
EF
.
U
se i
som
etri
c d
ot p
aper
an
d t
he
orth
ogra
ph
ic d
raw
ing
to s
ket
ch a
sol
id.
•
Th
e to
p vi
ew i
ndi
cate
s tw
o co
lum
ns.
•
Th
e ri
ght
and
left
vie
ws
indi
cate
th
at t
he
hei
ght
of f
igu
re i
s th
ree
bloc
ks.
•
Th
e fr
ont
view
in
dica
tes
that
th
e co
lum
ns
hav
e h
eigh
ts 2
an
d 3
bloc
ks.
Con
nec
t th
e do
ts o
n t
he
isom
etri
c do
t pa
per
to r
epre
sen
t th
e ed
ges
of t
he
soli
d. S
had
e th
e to
ps o
f ea
ch c
olu
mn
.
Exer
cise
sS
ket
ch e
ach
sol
id u
sin
g is
omet
ric
dot
pap
er.
1. c
ube
wit
h 4
un
its
on e
ach
sid
e 2.
rec
tan
gula
r pr
ism
1 u
nit
hig
h, 5
un
its
lon
g, a
nd
4 u
nit
s w
ide
Use
iso
met
ric
dot
pap
er a
nd
eac
h o
rth
ogra
ph
ic d
raw
ing
to s
ket
ch a
sol
id.
3.
top
view
left
view
front
vie
wrig
ht v
iew
4.
top
view
left
view
front
vie
wrig
ht v
iew
Stud
y G
uide
and
Inte
rven
tion
Rep
resen
tati
on
s o
f Th
ree-D
imen
sio
nal
Fig
ure
s
12-1
Exam
ple
1
A
BC
D
EF
Exam
ple
2
top
view
left
view
front
vie
wrig
ht v
iew
obje
ct
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pyrig
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cGraw
-Hill, a d
ivision o
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cGraw
-Hill C
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panies, Inc.
PDF Pass
Chapter 12 A2 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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ME
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ME
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TE
PE
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Cha
pte
r 12
6
G
lenc
oe G
eom
etry
Cro
ss S
ecti
on
s T
he
inte
rsec
tion
of
a so
lid
and
a pl
ane
is c
alle
d a
cros
s se
ctio
n o
f th
e so
lid.
Th
e sh
ape
of a
cro
ss s
ecti
on d
epen
ds u
pon
th
e an
gle
of t
he
plan
e.
Th
ere
are
seve
ral
inte
rest
ing
shap
es t
hat
are
cro
ss s
ecti
ons
of a
con
e. D
eter
min
e th
e sh
ape
resu
ltin
g fr
om e
ach
cro
ss s
ecti
on o
f th
e co
ne.
a.
If t
he
plan
e is
par
alle
l to
th
e ba
se o
f th
e co
ne,
then
th
e re
sult
ing
cros
s se
ctio
n w
ill
be a
circ
le.
b.
If t
he
plan
e cu
ts t
hro
ugh
th
e co
ne
perp
endi
cula
r to
th
e ba
se a
nd
thro
ugh
th
e ce
nte
r of
th
e co
ne,
th
en t
he
resu
ltin
g cr
oss
sect
ion
wil
l be
a t
rian
gle.
c.
If t
he
plan
e cu
ts a
cros
s th
e en
tire
con
e, t
hen
the
resu
ltin
g cr
oss
sect
ion
wil
l be
an
ell
ipse
.
Exer
cise
sD
escr
ibe
each
cro
ss s
ecti
on.
1.
2.
3.
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Rep
resen
tati
on
s o
f Th
ree-D
imen
sio
nal
Fig
ure
s
12-1
Ho
rizo
nta
l cro
ss s
ecti
on
Vert
ical
cro
ss
secti
on
An
gle
d c
ross
secti
on
Exam
ple c
ircle
ellip
se
recta
ng
le
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Lesson 12-1
Cha
pte
r 12
7
G
lenc
oe G
eom
etry
12-1
Use
iso
met
ric
dot
pap
er t
o sk
etch
eac
h p
rism
.
1. c
ube
2 u
nit
s on
eac
h e
dge
2. r
ecta
ngu
lar
pris
m 2
un
its
hig
h,
5
un
its
lon
g, a
nd
2 u
nit
s w
ide
Use
iso
met
ric
dot
pap
er a
nd
eac
h o
rth
ogra
ph
ic d
raw
ing
to s
ket
ch a
sol
id.
3.
4.
Des
crib
e ea
ch c
ross
sec
tion
.
5.
6.
7.
8.
recta
ng
le
recta
ng
lesq
uare
Skill
s Pr
acti
ceR
ep
resen
tati
on
s o
f Th
ree-D
imen
sio
nal
Fig
ure
s
top
view
left
view
front
vie
wrig
ht v
iew
tria
ng
le
top
view
left
view
front
vie
wrig
ht v
iew
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An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A3 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
8
G
lenc
oe G
eom
etry
Use
iso
met
ric
dot
pap
er t
o sk
etch
eac
h p
rism
.
1. r
ecta
ngu
lar
pris
m 3
un
its
hig
h,
2. t
rian
gula
r pr
ism
3 u
nit
s h
igh
, wh
ose
base
s 3
un
its
lon
g, a
nd
2 u
nit
s w
ide
are
rig
ht
tria
ngl
es w
ith
leg
s 2
un
its
and
4
un
its
lon
g
Use
iso
met
ric
dot
pap
er a
nd
eac
h o
rth
ogra
ph
ic d
raw
ing
to s
ket
ch a
sol
id.
3.
top
view
left
view
front
vie
wrig
ht v
iew
4.
top
view
left
view
front
vie
wrig
ht v
iew
Sk
etch
th
e cr
oss
sect
ion
fro
m a
ver
tica
l sl
ice
of e
ach
fig
ure
.
5.
6.
7. S
PHER
ES C
onsi
der
the
sph
ere
in E
xerc
ise
5. B
ased
on
th
e cr
oss
sect
ion
res
ult
ing
from
a
hor
izon
tal
and
a ve
rtic
al s
lice
of
the
sph
ere,
mak
e a
con
ject
ure
abo
ut
all
sph
eric
al
cros
s se
ctio
ns.
8. M
INER
ALS
Pyr
ite,
als
o kn
own
as
fool
’s g
old,
can
for
m c
ryst
als
that
are
per
fect
cu
bes.
S
upp
ose
a ge
mol
ogis
t w
ants
to
cut
a cu
be o
f py
rite
to
get
a sq
uar
e an
d a
rect
angl
ar f
ace.
W
hat
cu
ts s
hou
ld b
e m
ade
to g
et e
ach
of
the
shap
es?
Illu
stra
te y
our
answ
ers.
Prac
tice
Rep
resen
tati
on
s o
f Th
ree-D
imen
sio
nal
Fig
ure
s
12-1
cir
cle
trap
ezo
id
All s
ph
eri
cal
cro
ss s
ecti
on
s a
re c
ircle
s.
a c
ut
para
llel
to
the b
ases t
o g
et
a s
qu
are
a c
ut
thro
ug
h
dia
go
nally o
pp
osit
e
top
an
d b
ott
om
ed
ges
to g
et
a r
ecta
ng
le
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Lesson 12-1
Cha
pte
r 12
9
G
lenc
oe G
eom
etry
1. L
AB
ELS
Jam
al r
emov
es t
he
labe
l fr
om a
cy
lin
dric
al s
oup
can
to
earn
poi
nts
for
h
is s
choo
l. S
ketc
h t
he
shap
e of
th
e la
bel.
2. B
LOC
KS
Mar
got’s
th
ree-
year
-old
son
m
ade
the
mag
net
ic b
lock
scu
lptu
re
show
n b
elow
in
cor
ner
vie
w.
D
raw
th
e ri
ght
view
of
the
scu
lptu
re.
3. C
UB
ES N
ath
an m
arks
th
e m
idpo
ints
of
thre
e ed
ges
of a
cu
be a
s sh
own
.H
e th
en s
lice
s th
e cu
be a
lon
g a
plan
e th
at c
onta
ins
thes
e th
ree
poin
ts.
Des
crib
e th
e re
sult
ing
cros
s se
ctio
n.
4. E
NG
INEE
RIN
G S
teph
anie
nee
ds a
n
obje
ct w
hos
e to
p vi
ew i
s a
circ
le a
nd
wh
ose
left
an
d fr
ont
view
s ar
e sq
uar
es.
Des
crib
e an
obj
ect
that
wil
l sa
tisf
y th
ese
con
diti
ons.
5. D
ESK
SU
PPO
RTS
Th
e fi
gure
sh
ows
the
supp
ort
for
a de
sk.
a. D
raw
th
e to
p vi
ew.
b.
Dra
w t
he
fron
t vi
ew.
c. D
raw
th
e ri
ght
view
.
Wor
d Pr
oble
m P
ract
ice
Rep
resen
tati
on
s o
f Th
ree-D
imen
sio
nal
Fig
ure
s
12-1 reg
ula
r h
exag
on
Sam
ple
an
sw
er:
A c
ylin
der
wit
h
its h
eig
ht
eq
ual
to i
ts d
iam
ete
r.
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pyrig
ht © G
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ivision o
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cGraw
-Hill C
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Chapter 12 A4 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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ME
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Cha
pte
r 12
1
0
Gle
ncoe
Geo
met
ry
Dra
win
g S
olid
s o
n I
so
metr
ic D
ot
Pap
er
Isom
etri
c do
t pa
per
is h
elpf
ul
for
draw
ing
soli
ds. R
emem
ber
to u
se
dash
ed l
ines
for
hid
den
edg
es.
For
eac
h s
olid
sh
own
, dra
w a
not
her
sol
id w
hos
e d
imen
sion
s ar
e tw
ice
as l
arge
.
1.
2.
3.
4.
5.
6.
Enri
chm
ent
12-1
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Lesson 12-1
Cha
pte
r 12
1
1
Gle
ncoe
Geo
met
ry
Th
e sc
ien
ce o
f pe
rspe
ctiv
e dr
awin
g st
udi
es h
ow t
o dr
aw a
th
ree-
dim
ensi
onal
obj
ect
on a
tw
o-di
men
sion
al p
age.
Th
is s
cien
ce b
ecam
e h
igh
ly
refi
ned
du
rin
g th
e R
enai
ssan
ce w
ith
th
e w
ork
of a
rtis
ts s
uch
as
Alb
rech
t D
üre
r an
d L
eon
ardo
da
Vin
ci.
Tod
ay, c
ompu
ters
are
oft
en u
sed
to m
ake
pers
pect
ive
draw
ings
, par
ticu
larl
y el
abor
ate
grap
hic
s u
sed
in t
elev
isio
n a
nd
mov
ies.
Th
e th
ree-
dim
ensi
onal
co
ordi
nat
es o
f ob
ject
s ar
e fi
gure
d. T
hen
alg
ebra
is
use
d to
tra
nsf
orm
th
ese
into
tw
o-di
men
sion
al c
oord
inat
es. T
he
grap
h o
f th
ese
new
coo
rdin
ates
is
call
ed a
pro
ject
ion
.
Th
e fo
rmu
las
belo
w w
ill
draw
on
e ty
pe o
f pr
ojec
tion
in
wh
ich
th
e y-
axis
is
draw
n h
oriz
onta
lly,
th
e z-
axis
ver
tica
lly,
an
d th
e x-
axis
at
an a
ngl
e of
a˚
wit
h t
he
y-ax
is. I
f th
e th
ree-
dim
ensi
onal
coo
rdin
ates
of
a po
int
are
(x, y
, z),
th
en t
he
proj
ecti
on c
oord
inat
es (
X, Y
) ar
e gi
ven
by
X =
x(-
cos
a) +
y a
nd
Y =
x(-
sin
a)
+ z
.A
lth
ough
th
is t
ype
of p
roje
ctio
n g
ives
a f
airl
y go
od p
ersp
ecti
ve d
raw
ing,
it
does
dis
tort
som
e le
ngt
hs.
1. T
he
draw
ing
wit
h t
he
coor
din
ates
giv
en b
elow
is
a cu
be.
A
(5, 0
, 5),
B(5
, 5, 5
), C
(5, 5
, 0),
D(5
, 0, 0
),
E(0
, 0, 5
), F
(0, 5
, 5),
G(0
, 5, 0
), H
(0, 0
, 0)
U
se t
he
form
ula
s ab
ove
to f
ind
the
proj
ecti
on c
oord
inat
es o
f ea
ch
poin
t, u
sin
g a
= 4
5. R
oun
d pr
ojec
tion
coo
rdin
ates
to
the
nea
rest
in
tege
r. G
raph
th
e cu
be o
n a
gra
phin
g ca
lcu
lato
r. M
ake
a sk
etch
of
th
e di
spla
y.
A'(_
_, _
_) B
'(__,
__)
C′(_
_, _
_) D
′(__,
__)
E' (_
_, _
_) F
'(__,
__)
G′(_
_, _
_) H
′(__,
__)
2. T
he
poin
ts A
(10,
2, 0
), B
(10,
10,
0),
C(2
, 10,
0),
an
d D
(3, 3
, 4)
are
vert
ices
of
a py
ram
id.
Fin
d th
e pr
ojec
tion
coo
rdin
ates
, usi
ng
a =
25.
Rou
nd
coor
din
ates
to
the
nea
rest
in
tege
r.
Th
en g
raph
th
e py
ram
id o
n a
gra
phin
g ca
lcu
lato
r by
dra
win
g −
−−
A
′B′ ,
−−
−
B
′C′ ,
−−
−
C
′D′ ,
−−
−
D
′A′ ,
and
−−
−
D
′B′ .
Mak
e a
sket
ch o
f th
e di
spla
y.
A′(_
_, _
_) B
′(__,
__)
C′(_
_, _
_) D
′(__,
__)
A'(-
7,
-4),
B'(1
, -
4),
C'(8
, -
1),
D'(0
, 3);
fo
r sketc
hes,
see s
tud
en
ts’
wo
rk.
Gra
phin
g Ca
lcul
ator
Act
ivit
yP
ers
pecti
ve D
raw
ing
s
A'(-
4,
1),
B'(1
, 1),
C'(1
, -
4),
D'(-
4,
-4),
E'(0
, 5),
F'(5
, 5),
G'(5
, 0)
H'(0
,0);
F
or
sketc
hes,
see s
tud
en
ts’
wo
rk.
12-1
EF
AB
D
H
C
G
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Answers (Lesson 12-1)
A01-A26_GEOCRMC12_890521.indd A4A01-A26_GEOCRMC12_890521.indd A4 6/14/08 9:20:52 PM6/14/08 9:20:52 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A5 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
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RIO
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NA
ME
DA
TE
PE
RIO
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Cha
pte
r 12
1
2
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
Su
rface A
reas o
f P
rism
s a
nd
Cylin
ders
12-2
Late
ral a
nd
Su
rfac
e A
reas
of
Pris
ms
In a
sol
id f
igu
re,
face
s th
at a
re n
ot b
ases
are
lat
eral
fac
es. T
he
late
ral
area
is
the
sum
of
the
area
of
the
late
ral
face
s. T
he
surf
ace
area
is
the
sum
of
the
late
ral
area
an
d th
e ar
ea o
f th
e ba
ses.
La
tera
l A
rea
of
a P
ris
m
If a
prism
ha
s a
la
tera
l a
rea
of
L sq
ua
re u
nits,
a h
eig
ht
of
h u
nits,
an
d e
ach
ba
se
ha
s a
pe
rim
ete
r o
f P
un
its,
the
n L
= P
h.
Su
rfa
ce
Are
a
of
a P
ris
m
If a
prism
ha
s a
su
rfa
ce
are
a o
f S
sq
ua
re u
nits,
a la
tera
l a
rea
of
L sq
ua
re u
nits,
an
d e
ach
ba
se
ha
s a
n a
rea
of
B s
qu
are
un
its,
the
n
S =
L +
2B
or
S =
Ph
+ 2
B
F
ind
th
e la
tera
l an
d s
urf
ace
area
of
the
regu
lar
pen
tago
nal
p
rism
ab
ove
if e
ach
bas
e h
as a
per
imet
er o
f 75
cen
tim
eter
s an
d t
he
hei
ght
is
10 c
enti
met
ers.
L =
Ph
Late
ral are
a o
f a p
rism
=
75(
10)
P =
75,
h =
10
=
750
M
ultip
ly.
Th
e la
tera
l ar
ea i
s 75
0 sq
uar
e ce
nti
met
ers
and
the
surf
ace
area
is
abou
t 15
24.2
squ
are
cen
tim
eter
s.
Exer
cise
sF
ind
th
e la
tera
l ar
ea a
nd
su
rfac
e ar
ea o
f ea
ch p
rism
. Rou
nd
to
the
nea
rest
ten
th
if n
eces
sary
.
1.
4 m
3 m
10 m
2.
15 in
.
10 in
.
8 in
.
3.
6 in
.18
in.
4.
20 c
m
10 c
m10
cm
12 c
m8
cm9
cm
5.
4 in
.
4 in
.
12 in
.
6.
4 m
16 m
Exam
ple
pent
agon
al p
rismal
titud
e
late
ral
edge
late
ral
face
S =
L +
2B
=
750
+ 2
( 1 −
2 a
P)
=
750
+ (
7.5
−
tan
36°
) (75)
≈
152
4.2
tan
36°
= 7.
5 −
a
a
=
7.5
−
tan
36°
a 15 c
m36°
L =
120 m
2; S
= 1
32 m
2
L =
540 i
n2 (
10 i
n.
× 8 i
n.
base);
S
= 7
00 i
n2
L =
460 i
n2
(8 i
n.
× 1
5 i
n.
base)
or
L =
400 i
n2
(10 in
. ×
15 in
. b
ase)
or
L =
540 i
n2; S
≈ 6
63.9
in
2
L =
588 c
m2; S
= 8
28 c
m2
L =
128 in
2 (
recta
ng
ula
r b
ase)
or
L =
384 m
2; S
= 4
67.1
m2
L =
192 i
n2 (
sq
uare
base);
S =
224 i
n2
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ME
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Lesson 12-2
Cha
pte
r 12
13
Gle
ncoe
Geo
met
ry
Late
ral a
nd
Su
rfac
e A
reas
of
Cyl
ind
ers
A c
ylin
der
is
a so
lid
wit
h b
ases
th
at a
re c
ongr
uen
t ci
rcle
s ly
ing
in p
aral
lel
plan
es.
Th
e ax
is o
f a
cyli
nde
r is
th
e se
gmen
t w
ith
en
dpoi
nts
at
the
cen
ters
of
th
ese
circ
les.
For
a r
igh
t cy
lin
der
, th
e ax
is i
s al
so t
he
alti
tude
of
th
e cy
lin
der.
La
tera
l A
rea
of
a C
yli
nd
er
If a
cylin
de
r h
as a
la
tera
l a
rea
of
L sq
ua
re u
nits,
a h
eig
ht
of
h u
nits,
an
d a
ba
se
ha
s a
ra
diu
s o
f r
un
its,
the
n L
= 2
πrh
.
Su
rfa
ce
Are
a
of
a C
yli
nd
er
If a
cylin
de
r h
as a
su
rfa
ce
are
a o
f S
sq
ua
re u
nits,
a h
eig
ht
of
h u
nits,
an
d a
ba
se
ha
s a
ra
diu
s o
f r
un
its,
the
n S
= L
+ 2
B o
r 2
πrh
+ 2
πr2
.
F
ind
th
e la
tera
l an
d s
urf
ace
area
of
the
cyli
nd
er. R
oun
d t
o th
e n
eare
st t
enth
.If
d =
12
cm, t
hen
r =
6 c
m.
L =
2πrh
Late
ral are
a o
f a c
ylin
der
=
2π(6
)(14
) r
= 6
, h
= 1
4
≈
527
.8
Use a
calc
ula
tor.
S =
2πrh
+ 2
πr2
Surf
ace a
rea o
f a c
ylin
der
≈
527
.8 +
2π(6
)2 2
πrh
≈ 5
27.8
, r
= 6
≈
754
.0
Use a
calc
ula
tor.
Th
e la
tera
l ar
ea i
s ab
out
527.
8 sq
uar
e ce
nti
met
ers
and
the
surf
ace
area
is
abou
t 75
4.0
squ
are
cen
tim
eter
s.
Exer
cise
sF
ind
th
e la
tera
l ar
ea a
nd
su
rfac
e ar
ea o
f ea
ch c
ylin
der
. Rou
nd
to
the
nea
rest
te
nth
. 1
.
12 c
m
4 cm
2.
6 in
.10
in.
3.
6 cm
3 cm
3 cm
4.
20 c
m
8 cm
5.
12 m
4 m
6.
2
m
1 m
radi
us o
f bas
eax
is
base
base
heig
ht
Exam
ple
14 c
m
12 c
m
12-2
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Su
rface A
reas o
f P
rism
s a
nd
Cylin
ders
L ≈
301.6
cm
2; S
≈ 4
02.1
cm
2L ≈
377.0
in
2; S
≈ 6
03.2
in
2
L ≈
113.1
cm
2; S
≈ 1
69.6
cm
2L ≈
502.7
cm
2; S
≈ 6
03.2
cm
2
L ≈
150.8
m2; S
≈ 3
77.0
m2
L ≈
12.6
m2; S
≈ 3
7.7
m2
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Answers (Lesson 12-2)
A01-A26_GEOCRMC12_890521.indd A5A01-A26_GEOCRMC12_890521.indd A5 6/2/09 5:31:16 PM6/2/09 5:31:16 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A6 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
1
4
Gle
ncoe
Geo
met
ry
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
pri
sm. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
1.
2.
3.
4.
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
cyl
ind
er. R
oun
d t
o th
e n
eare
st
ten
th.
5.
6.
7.
8.
12 y
d
12 y
d
10 y
d8
m
6 m
12 m
10 in
.5
in.
6 in
.
8 in
.
9 cm
9 cm7.8
cm9
cm
12 c
m
L
= 480 y
d2 (
sq
uare
base)
L =
240 m
2 (
8 ×
12 b
ase)
L =
528 y
d2 (
recta
ng
ula
r b
ase)
L =
288 m
2 (
12 ×
6 b
ase)
S =
768 y
d2
L =
336 m
2 (
8 ×
6 b
ase)
S =
432 m
2
L
= 120 i
n2
L =
324 c
m2
S
= 168 i
n2
S =
394.2
cm
2
Skill
s Pr
acti
ceS
urf
ace A
reas o
f P
rism
s a
nd
Cylin
ders
12-2
L
≈ 377.0
in
2
L ≈
25.1
m2
S
≈ 603.2
in
2
S ≈
50.3
m2
L
≈ 37.7
yd
2
L ≈
603.2
in
2
S
≈ 94.2
yd
2
S ≈
1005.3
in
2
8 in
.
12 in
.
2 yd
3 yd12
in.
10 in
.
2 m
2 m
001_
025_
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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Lesson 12-2
Cha
pte
r 12
15
Gle
ncoe
Geo
met
ry
12-2
Prac
tice
Su
rface A
reas o
f P
rism
s a
nd
Cylin
ders
Fin
d t
he
late
ral
and
su
rfac
e ar
ea o
f ea
ch p
rism
. Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
. 1
.
15 c
m
15 c
m
32 c
m
2.
8 ft
10 ft
5 ft
3.
2 m
11 m
4.
4 yd
4 yd
9.5
yd
5 yd
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
cyl
ind
er. R
oun
d t
o th
e n
eare
st
ten
th.
5.
5 ft 7
ft
6.
4 m
8.5
m
7.
19 in
.
17 in
. 8.
12 m
30 m
L =
1920 c
m2 (
sq
uare
base)
or
L =
224.3
ft2
;L =
1410 c
m2 (
recta
ng
ula
r b
ase);
S =
264.3
ft2
S =
2370 c
m2
L =
132 m
2; S
≈ 152.8
m2
L =
123.5
yd
2; S
≈ 1
39.1
yd
2
L ≈
219.9
ft2
;
L ≈
106.8
m2;
S ≈
377.0
ft2
S
≈ 131.9
m2
L ≈
1014.7
in
2;
L ≈
2261.9
m2;
S ≈
1581.8
in
2
S ≈
3166.7
m2
001_
025_
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OC
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8:32
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Answers (Lesson 12-2)
A01-A26_GEOCRMC12_890521.indd A6A01-A26_GEOCRMC12_890521.indd A6 3/5/13 10:27:21 PM3/5/13 10:27:21 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A7 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
1
6
Gle
ncoe
Geo
met
ry
1. L
OG
OS
Th
e Z
com
pan
y sp
ecia
lize
s in
ca
rin
g fo
r ze
bras
. Th
ey w
ant
to m
ake
a 3-
dim
ensi
onal
“Z
” to
pu
t in
fro
nt
of t
hei
r co
mpa
ny
hea
dqu
arte
rs. T
he
“Z”
is
15 i
nch
es t
hic
k an
d th
e pe
rim
eter
of
the
base
is
390
inch
es.
15"
W
hat
is
the
late
ral
surf
ace
area
of
this
“Z
”?
2. S
TAIR
WEL
LS M
anag
emen
t de
cide
s to
en
clos
e st
airs
con
nec
tin
g th
e fi
rst
and
seco
nd
floo
rs o
f a
park
ing
gara
ge i
n a
st
airw
ell
shap
ed l
ike
an o
bliq
ue
rect
angu
lar
pris
m.
16
ft
20
ft
15
ft
9 ft
W
hat
is
the
late
ral
surf
ace
area
of
the
stai
rwel
l?
3. C
AK
ES A
cak
e is
a r
ecta
ngu
lar
pris
m
wit
h h
eigh
t 4
inch
es a
nd
base
12
inch
es
by 1
5 in
ches
. Wal
lace
wan
ts t
o ap
ply
fros
tin
g to
th
e si
des
and
the
top
of t
he
cake
. Wh
at i
s th
e su
rfac
e ar
ea o
f th
e pa
rt o
f th
e ca
ke t
hat
wil
l h
ave
fros
tin
g?
4. E
XH
AU
ST P
IPES
An
exh
aust
pip
e is
sh
aped
lik
e a
cyli
nde
r w
ith
a h
eigh
t of
50
in
ches
an
d a
radi
us
of 2
in
ches
. Wh
at
is t
he
late
ral
surf
ace
area
of
the
exh
aust
pi
pe?
Rou
nd
you
r an
swer
to
the
nea
rest
h
un
dred
th.
5. T
OW
ERS
A c
ircu
lar
tow
er i
s m
ade
by
plac
ing
one
cyli
nde
r on
top
of
anot
her
. B
oth
cyl
inde
rs h
ave
a h
eigh
t of
18
inch
es. T
he
top
cyli
nde
r h
as a
rad
ius
of
18 i
nch
es a
nd
the
bott
om c
ylin
der
has
a
radi
us
of 3
6 in
ches
.
18
in.
18
in.
a. W
hat
is
the
tota
l su
rfac
e ar
ea o
f th
e to
wer
? R
oun
d yo
ur
answ
er t
o th
e n
eare
st h
un
dred
th.
b.
An
oth
er t
ower
is
con
stru
cted
by
plac
ing
the
orig
inal
tow
er o
n t
op o
f an
oth
er c
ylin
der
wit
h a
hei
ght
of
18 i
nch
es a
nd
a ra
diu
s of
54
inch
es.
Wh
at i
s th
e to
tal
surf
ace
area
of
the
new
tow
er?
Rou
nd
you
r an
swer
to
the
nea
rest
hu
ndr
edth
.
12-2
Wor
d Pr
oble
m P
ract
ice
Su
rface A
reas o
f P
rism
s a
nd
Cylin
ders
5850 i
n2
840 f
t2
396 i
n2
628.3
2 i
n2
14,2
50.2
6 i
n2
30,5
36.2
8 i
n2
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Lesson 12-2
Cha
pte
r 12
17
Gle
ncoe
Geo
met
ry
Enri
chm
ent
12-2
Min
imiz
ing
Co
st
in M
an
ufa
ctu
rin
gS
upp
ose
that
a m
anu
fact
ure
r w
ants
to
mak
e a
can
th
at h
as a
vol
um
e of
40
cu
bic
inch
es. T
he
cost
to
mak
e th
e ca
n i
s 3
cen
ts p
er s
quar
e in
ch f
or
the
top
and
bott
om a
nd
1 ce
nt
per
squ
are
inch
for
th
e si
de.
1. W
rite
th
e va
lue
of h
in
ter
ms
of r
, giv
en v
= π
r2h
.
h =
40
−
πr2
2. W
rite
a f
orm
ula
for
th
e co
st i
n t
erm
s of
r.
C
= 3
(2π
r2)
+ 1
(2π
r � 4
0
−
πr2
) o
r 6
πr2
+ 8
0
−
r
3. U
se a
gra
phin
g ca
lcu
lato
r to
gra
ph t
he
form
ula
, let
tin
g Y
1 re
pres
ent
the
cost
an
d X
re
pres
ent
r. U
se t
he
grap
h t
o es
tim
ate
the
poin
t at
wh
ich
th
e co
st i
s m
inim
ized
.
Th
e m
inim
um
heig
ht
is 7
.71 i
nch
es,
wh
ich
giv
es a
min
imu
m c
ost
of
93.
4. R
epea
t th
e pr
oced
ure
usi
ng
2 ce
nts
per
squ
are
inch
for
th
e to
p an
d bo
ttom
an
d 4
cen
ts p
er s
quar
e in
ch f
or t
he
top
and
bott
om.
2
cen
ts:
C =
4π
r2 +
80
−
r =
5.8
8 i
n.
; 4 c
en
ts:
C =
8π
r2 +
80
−
r =
9.3
4 i
n.
5. W
hat
wou
ld y
ou e
xpec
t to
hap
pen
as
the
cost
of
the
top
and
bott
om i
ncr
ease
s?
S
ee s
tud
en
ts’
wo
rk.
Sam
ple
an
sw
er:
Th
e m
an
ufa
ctu
rer
mig
ht
make t
he
can
s t
aller
an
d n
arr
ow
er.
6. C
ompu
te t
he
tabl
e fo
r th
e co
st v
alu
e gi
ven
. Wh
at h
appe
ns
to t
he
hei
ght
of t
he
can
as
the
cost
of
the
top
and
bott
om i
ncr
ease
s?
T
he h
eig
ht
incre
ases a
s t
he c
ost
of
the t
op
an
d b
ott
om
go
up
.
Co
st
To
pC
os
tM
inim
um
& B
ott
om
Cy
lin
de
rh
2 ce
nts
1 ce
nt
5.8
8
3 ce
nts
1 ce
nt
7.7
1
4 ce
nts
1 ce
nt
9.3
4
5 ce
nts
1 ce
nt
10.8
4
6 ce
nts
1 ce
nt
12.2
4
r
h
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Answers (Lesson 12-2)
A01-A26_GEOCRMC12_890521.indd A7A01-A26_GEOCRMC12_890521.indd A7 6/2/09 5:31:47 PM6/2/09 5:31:47 PM
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pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A8 Glencoe Geometry
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ME
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Cha
pte
r 12
1
8
Gle
ncoe
Geo
met
ry
12-3
Stud
y G
uide
and
Inte
rven
tion
Su
rface A
reas o
f P
yra
mid
s a
nd
Co
nes
Late
ral a
nd
Su
rfac
e A
reas
of
Pyra
mid
s A
pyr
amid
is
a so
lid
wit
h a
pol
ygon
bas
e. T
he
late
ral
face
s in
ters
ect
in a
com
mon
po
int
know
n a
s th
e ve
rtex
. Th
e al
titu
de i
s th
e se
gmen
t fr
om t
he
vert
ex t
hat
is
perp
endi
cula
r to
th
e ba
se. F
or a
reg
ula
r p
yram
id,
the
base
is
a re
gula
r po
lygo
n a
nd
the
alti
tude
has
an
en
dpoi
nt
at
the
cen
ter
of t
he
base
. All
th
e la
tera
l ed
ges
are
con
gru
ent
and
all
the
late
ral
face
s ar
e co
ngr
uen
t is
osce
les
tria
ngl
es. T
he
hei
ght
of e
ach
la
tera
l fa
ce i
s ca
lled
th
e sl
ant
hei
ght.
La
tera
l A
rea
of
a R
eg
ula
r P
yra
mid
Th
e la
tera
l a
rea
L o
f a
re
gu
lar
pyra
mid
is L
= 1
−
2 P
ℓ, w
he
re ℓ
is t
he
sla
nt
he
igh
t a
nd
P is t
he
pe
rim
ete
r o
f th
e b
ase
.
Su
rfa
ce
Are
a o
f
a R
eg
ula
r P
yra
mid
Th
e s
urf
ace
are
a S
of
a r
eg
ula
r p
yra
mid
is S
=
1
−
2 P
ℓ +
B,
wh
ere
ℓ is t
he
sla
nt
he
igh
t, P
is t
he
pe
rim
ete
r o
f th
e b
ase
,
an
d B
is t
he
are
a o
f th
e b
ase
.
F
or t
he
regu
lar
squ
are
pyr
amid
ab
ove,
fin
d t
he
late
ral
area
an
d
surf
ace
area
if
the
len
gth
of
a si
de
of t
he
bas
e is
12
cen
tim
eter
s an
d t
he
hei
ght
is
8 ce
nti
met
ers.
Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
Fin
d th
e sl
ant
hei
ght.
ℓ2 =
62
+ 8
2 P
yth
agore
an T
heore
m
ℓ2 =
100
S
implif
y.
ℓ =
10
Take t
he p
ositiv
e s
quare
root
of
each s
ide.
Th
e la
tera
l ar
ea i
s 24
0 sq
uar
e ce
nti
met
ers,
an
d th
e su
rfac
e ar
ea i
s 38
4 sq
uar
e ce
nti
met
ers.
Exer
cise
sF
ind
th
e la
tera
l ar
ea a
nd
su
rfac
e ar
ea o
f ea
ch r
egu
lar
pyr
amid
. Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
1.
15 c
m
20 c
m
2.
45°
8 ft
3.
60°
10 c
m
4.
6 in
.8.
7 in
.15
in.
late
ral e
dge
base
slan
t hei
ght
heig
ht
Exam
ple
L =
1 −
2 P
ℓ Late
ral are
a o
f a r
egula
r pyra
mid
=
1 −
2 (4
8)(1
0)
P =
4 �
12
or
48, ℓ
= 1
0
=
240
S
implif
y.
S =
1 −
2 P
ℓ +
B
Surf
ace a
rea o
f a r
egula
r pyra
mid
=
240
+ 1
44
1
−
2 P
ℓ =
240, B
= 1
2 ·
12 o
r 144
=
384
L ≈
450 c
m2; S
≈ 5
47.4
cm
2
L ≈
362.0
ft2
; S ≈
618.0
ft2
L ≈
266.7
cm
2; S
≈ 4
00.0
cm
2
L ≈
326.3
in
2; S
≈ 4
56.8
in
2
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Lesson 12-3
Cha
pte
r 12
19
Gle
ncoe
Geo
met
ry
12-3
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Su
rface A
reas o
f P
yra
mid
s a
nd
Co
nes
Late
ral a
nd
Su
rfac
e A
reas
of
Co
nes
A c
one
has
a
circ
ula
r ba
se a
nd
a ve
rtex
. Th
e ax
is o
f th
e co
ne
is t
he
segm
ent
wit
h e
ndp
oin
ts a
t th
e ve
rtex
an
d th
e ce
nte
r of
th
e ba
se. I
f th
e ax
is i
s al
so t
he
alti
tude
, th
en t
he
con
e is
a
righ
t co
ne.
If
the
axis
is
not
th
e al
titu
de, t
hen
th
e co
ne
is a
n o
bli
qu
e co
ne.
La
tera
l A
rea
of
a C
on
e
Th
e la
tera
l a
rea
L o
f a
rig
ht
circu
lar
co
ne
is L
= π
r�,
wh
ere
r is
the
ra
diu
s a
nd
� is t
he
sla
nt
he
igh
t.
Su
rfa
ce
Are
a o
f
a C
on
e
Th
e s
urf
ace
are
a S
of
a r
igh
t co
ne
is S
= π
r� +
πr2
, w
he
re r
is
the
ra
diu
s a
nd
� is t
he
sla
nt
he
igh
t.
F
or t
he
righ
t co
ne
abov
e, f
ind
th
e la
tera
l ar
ea a
nd
su
rfac
e ar
ea i
f th
e ra
diu
s is
6 c
enti
met
ers
and
th
e h
eigh
t is
8 c
enti
met
ers.
Rou
nd
to
the
nea
rest
te
nth
if
nec
essa
ry.
Fin
d th
e sl
ant
hei
ght.
ℓ2 =
62
+ 8
2 P
yth
agore
an T
heore
m
ℓ2 =
100
S
implif
y.
ℓ =
10
Take t
he p
ositiv
e s
quare
root
of
each s
ide.
Th
e la
tera
l ar
ea i
s ab
out
188.
5 sq
uar
e ce
nti
met
ers
and
the
surf
ace
area
is
abou
t 30
1.6
squ
are
cen
tim
eter
s.
Exer
cise
sF
ind
th
e la
tera
l ar
ea a
nd
su
rfac
e ar
ea o
f ea
ch c
one.
Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
1.
9 cm
12 c
m
2.
5 ft
30°
3.
12 c
m
13 c
m
4.
4 in
.
45°
axis
base
base
�sl
ant h
eigh
t
right
con
eob
lique
con
e
altit
ude
VV
Exam
ple
L =
πrℓ
Late
ral are
a o
f a r
ight
cone
=
π(6
)(10
) r
= 6
, ℓ
= 1
0
≈
188
.5
Sim
plif
y.
S =
πrℓ
+ π
r2 S
urf
ace a
rea o
f a r
ight
cone
≈
188
.5 +
π(6
2 )
πrℓ
≈ 1
88.5
, r
= 6
≈
301
.6
Sim
plif
y.
L ≈
424.1
cm
2;
L ≈
157.1
ft2
;
S ≈
678.6
cm
2
S
≈ 2
35.6
ft2
L ≈
204.2
cm
2;
L ≈
71.1
in
2;
S ≈
282.7
cm
2
S
≈ 1
21.4
in
2
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Answers (Lesson 12-3)
A01-A26_GEOCRMC12_890521.indd A8A01-A26_GEOCRMC12_890521.indd A8 3/27/10 10:55:24 PM3/27/10 10:55:24 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
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c.
PDF Pass
Chapter 12 A9 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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ME
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Cha
pte
r 12
2
0
Gle
ncoe
Geo
met
ry
12-3
Skill
s Pr
acti
ceS
urf
ace A
reas o
f P
yra
mid
s a
nd
Co
nes
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
reg
ula
r p
yram
id. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
1.
2.
3.
4.
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
con
e. R
oun
d t
o th
e n
eare
st t
enth
.
5.
6.
7.
8.
4 cm
7 cm
20 in
.
8 in
.
9 m 10
m14
ft
12 ft
L
= 5
6 c
m2
L =
480 i
n2
S
= 7
2 c
m2
S =
646.3
in
2
L
≈ 283.2
m2
L ≈
389.0
ft2
S
≈ 4
55.3
m2
S ≈
585.0
ft2
14 m
5 m
10 ft
25 ft
8 in
.
21 in
.
17 m
m9 m
m
L
≈ 219.9
m2
L ≈
845.9
ft2
S
≈ 2
98.5
m2
S ≈
1160.1
ft2
L
≈ 5
27.8
in
2
L ≈
480.7
mm
2
S
≈ 7
28.8
in
2
S ≈
735.1
mm
2
001_
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D
Lesson 12-3
Cha
pte
r 12
21
Gle
ncoe
Geo
met
ry
12-3
Prac
tice
S
urf
ace A
reas o
f P
yra
mid
s a
nd
Co
nes
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
reg
ula
r p
yram
id. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
1.
9 yd
10 y
d
2.
12 m
7 m
3.
13 ft
5 ft
4.
8 cm
2.5
cm
Fin
d t
he
late
ral
area
an
d s
urf
ace
area
of
each
con
e. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
5.
5 m
4 m
6.
7 cm
21 c
m
7. F
ind
the
surf
ace
area
of
a co
ne
if t
he
hei
ght
is 1
4 ce
nti
met
ers
and
the
slan
t h
eigh
t is
16
.4 c
enti
met
ers.
8. F
ind
the
surf
ace
area
of
a co
ne
if t
he
hei
ght
is 1
2 in
ches
an
d th
e di
amet
er i
s 27
in
ches
.
9. G
AZE
BO
S T
he
roof
of
a ga
zebo
is
a re
gula
r oc
tago
nal
pyr
amid
. If
the
base
of
the
pyra
mid
has
sid
es o
f 0.
5 m
eter
an
d th
e sl
ant
hei
ght
of t
he
roof
is
1.9
met
ers,
fin
d th
e ar
ea o
f th
e ro
of.
10. H
ATS
Cuo
ng b
ough
t a
coni
cal
hat
on a
rec
ent
trip
to
cent
ral
Vie
tnam
. The
bas
ic f
ram
e of
th
e h
at i
s 16
hoo
ps o
f ba
mbo
o th
at g
radu
ally
dim
inis
h i
n s
ize.
Th
e h
at i
s co
vere
d in
pa
lm l
eave
s. I
f th
e h
at h
as a
dia
met
er o
f 50
cen
tim
eter
s an
d a
slan
t h
eigh
t of
32
cen
tim
eter
s, w
hat
is
the
late
ral
area
of
the
con
ical
hat
?
L =
180 y
d2;
S =
261 y
d2
L =
126 m
2; S
≈ 147.2
m2
L =
162.5
ft2
; S
≈ 2
05.5
ft2
L ≈
60 c
m2;
S ≈
76.2
cm
2
L ≈
80.5
m2;
S ≈
130.7
cm
2
468.8
cm
2 ; S
≈ 640.7
cm
2
669.3
cm
2
1338.6
in
2
3.8
m2
ab
ou
t 2513.3
cm
2
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Answers (Lesson 12-3)
A01-A26_GEOCRMC12_890521.indd A9A01-A26_GEOCRMC12_890521.indd A9 6/1/09 12:11:24 PM6/1/09 12:11:24 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A10 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
2
2
Gle
ncoe
Geo
met
ry
12-3
Wor
d Pr
oble
m P
ract
ice
Su
rface A
reas o
f P
yra
mid
s a
nd
Co
nes
1. P
APE
R M
OD
ELS
Pat
rick
is
mak
ing
a pa
per
mod
el
of a
cas
tle.
Par
t of
th
e m
odel
in
volv
es
cutt
ing
out
the
net
sh
own
an
d fo
ldin
g it
in
to a
py
ram
id. T
he
pyra
mid
has
a s
quar
e ba
se. W
hat
is
the
late
ral
surf
ace
area
of
the
resu
ltin
g py
ram
id?
2. T
ETR
AH
EDR
ON
Su
ng
Li
buil
ds a
pap
er
mod
el o
f a
regu
lar
tetr
ahed
ron
, a
pyra
mid
wit
h a
n e
quil
ater
al t
rian
gle
for
the
base
an
d th
ree
equ
ilat
eral
tri
angl
es
for
the
late
ral
face
s. O
ne
of t
he
face
s of
th
e te
trah
edro
n h
as a
n a
rea
of 1
7 sq
uar
e in
ches
. Wh
at i
s th
e to
tal
surf
ace
area
of
the
tetr
ahed
ron
?
3. P
APE
RW
EIG
HTS
Dap
hn
e u
ses
a pa
perw
eigh
t sh
aped
lik
e a
pyra
mid
wit
h
a re
gula
r h
exag
on f
or a
bas
e. T
he
side
le
ngt
h o
f th
e re
gula
r h
exag
on i
s 1
inch
. T
he
alti
tude
of
the
pyra
mid
is
2 in
ches
.
W
hat
is
the
late
ral
surf
ace
area
of
this
py
ram
id?
Rou
nd
you
r an
swer
s to
th
e n
eare
st h
un
dred
th.
4. S
PRA
Y P
AIN
T A
can
of
spra
y pa
int
shoo
ts o
ut
pain
t in
a c
one
shap
ed m
ist.
T
he
late
ral
surf
ace
area
of
the
con
e is
65
π sq
uar
e in
ches
wh
en t
he
can
is
hel
d 12
in
ches
fro
m a
can
vas.
Wh
at i
s th
e ar
ea o
f th
e pa
rt o
f th
e ca
nva
s th
at g
ets
spra
yed
wit
h p
ain
t? R
oun
d yo
ur
answ
er
to t
he
nea
rest
hu
ndr
edth
.
5. M
EGA
PHO
NES
A m
egap
hon
e is
fo
rmed
by
taki
ng
a co
ne
wit
h a
rad
ius
of 2
0 ce
nti
met
ers
and
an a
ltit
ude
of
60 c
enti
met
ers
and
cutt
ing
off
the
tip.
T
he
cut
is m
ade
alon
g a
plan
e th
at i
s pe
rpen
dicu
lar
to t
he
axis
of
the
con
e an
d in
ters
ects
th
e ax
is 1
2 ce
nti
met
ers
from
th
e ve
rtex
. Rou
nd
you
r an
swer
s to
th
e n
eare
st h
un
dred
th.
a. W
hat
is
the
late
ral
surf
ace
area
of
the
orig
inal
con
e?
b.
Wh
at i
s th
e la
tera
l su
rfac
e ar
ea o
f th
e ti
p th
at i
s re
mov
ed?
c. W
hat
is
the
late
ral
surf
ace
area
of
the
meg
aph
one?
20 c
m20
cm
15 c
m
600 c
m2
68 i
n2
6.5
4 i
n2
3973.8
4 i
n2
158.9
5 i
n2
3814.8
9 i
n2
78.5
4 i
n2
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Lesson 12-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
23
Gle
ncoe
Geo
met
ry
Co
ne P
att
ern
s
Th
e p
atte
rn a
t th
e ri
ght
is m
ade
from
a
circ
le. I
t ca
n b
e fo
lded
to
mak
e a
con
e.
1. M
easu
re t
he
radi
us
of t
he
circ
le t
o th
e n
eare
st c
enti
met
er.
2. T
he
patt
ern
is
wh
at f
ract
ion
of
the
com
plet
e ci
rcle
?
3. W
hat
is
the
circ
um
fere
nce
of
the
com
plet
e ci
rcle
?
4. H
ow l
ong
is t
he
circ
ula
r ar
c th
at i
s th
e ou
tsid
e of
th
e pa
tter
n?
5. C
ut
out
the
patt
ern
an
d ta
pe i
t to
geth
er t
o fo
rm a
con
e.
6. M
easu
re t
he
diam
eter
of
the
circ
ula
r ba
se o
f th
e co
ne.
7. W
hat
is
the
circ
um
fere
nce
of
the
base
of
the
con
e?
8. W
hat
is
the
slan
t h
eigh
t of
th
e co
ne?
9. U
se t
he
Pyt
hag
orea
n T
heo
rem
to
calc
ula
te t
he
hei
ght
of t
he
con
e.
Use
a d
ecim
al a
ppro
xim
atio
n. C
hec
k yo
ur
calc
ula
tion
by
mea
suri
ng
the
hei
ght
wit
h a
met
ric
rule
r.
10. F
ind
the
late
ral
area
.
11. F
ind
the
tota
l su
rfac
e ar
ea.
Mak
e a
pap
er p
atte
rn f
or e
ach
con
e w
ith
th
e gi
ven
mea
sure
men
ts.
Th
en c
ut
the
pat
tern
ou
t an
d m
ake
the
con
e. F
ind
th
e m
easu
rem
ents
.
12.
120°
6 cm
13
.
20 c
m
d
iam
eter
of
base
=
dia
met
er o
f ba
se =
l
ater
al a
rea
=
lat
eral
are
a =
h
eigh
t of
con
e =
h
eigh
t of
con
e =
(to
nea
rest
ten
th o
f a
cen
tim
eter
) (
to n
eare
st t
enth
of
a ce
nti
met
er)
Enri
chm
ent
12-3
2 c
m
3
−
4
4π
cm
3π
cm
See s
tud
en
ts’
wo
rk.
3π
cm
2 c
m
3 c
m
1.3
2 c
m
3π
cm
2
5.2
5 c
m2
8 c
m10 c
m
24
π c
m2
50
π c
m2
4.5
cm
8.7
cm
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Answers (Lesson 12-3)
A01-A26_GEOCRMC12_890521.indd A10A01-A26_GEOCRMC12_890521.indd A10 6/1/09 12:11:30 PM6/1/09 12:11:30 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A11 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
2
4
Gle
ncoe
Geo
met
ry
12-3
You
can
use
a s
prea
dsh
eet
to d
eter
min
e th
e su
rfac
e ar
ea o
f a
con
e.
L
ucy
wan
ts t
o w
rap
a M
oth
er’s
Day
gif
t. T
he
gift
sh
e h
as b
ough
t fo
r h
er m
oth
er i
s in
a c
onic
al b
ox t
hat
has
a s
lan
t h
eigh
t of
6 i
nch
es a
nd
has
a r
adiu
s of
3 i
nch
es. S
he
mu
st d
eter
min
e th
e su
rfac
e ar
ea o
f th
e b
ox t
o d
eter
min
e h
ow
mu
ch w
rap
pin
g p
aper
to
bu
y. U
se a
sp
read
shee
t to
det
erm
ine
the
surf
ace
area
of
the
box
. Rou
nd
to
the
nea
rest
ten
th.
Ste
p 1
U
se c
ell
A1
for
the
radi
us
of t
he
con
e an
d ce
ll B
1 fo
r th
e h
eigh
t.
Ste
p 2
In
cel
l C
1, e
nte
r an
equ
als
sign
fol
low
ed b
y P
I()*
A1*
B1
+ P
I()*
A1^
2. T
hen
pre
ss
EN
TE
R. T
his
wil
l re
turn
th
e su
rfac
e ar
ea o
f th
e co
ne.
Th
e su
rfac
e ar
ea o
f th
e co
nic
al b
ox i
s 84
.8 i
n2
to t
he
nea
rest
ten
th.
U
se a
sp
read
shee
t to
det
erm
ine
the
surf
ace
area
of
a co
ne
that
has
a r
adiu
s of
2.
5 ce
nti
met
ers
and
a s
lan
t h
eigh
t of
5.2
cen
tim
eter
s.
Rou
nd
to
the
nea
rest
ten
th.
Ste
p 1
U
se c
ell
A2
for
the
radi
us
of t
he
con
e an
d ce
ll B
2 fo
r th
e sl
ant
hei
ght.
Ste
p 2
C
lick
on
th
e bo
ttom
rig
ht
corn
er o
f ce
ll C
1 an
d dr
ag i
t to
C2.
Th
is r
etu
rns
the
surf
ace
area
of
the
con
e.
Th
e su
rfac
e ar
ea o
f th
e co
ne
is 6
0.5
cm2
to t
he
nea
rest
ten
th.
Exer
cise
sU
se a
sp
read
shee
t to
fin
d t
he
surf
ace
area
of
each
con
e w
ith
th
e gi
ven
d
imen
sion
s. R
oun
d t
o th
e n
eare
st t
enth
.
1. r
= 1
2 m
, � =
2.3
m
2.
r =
6 m
, � =
2 m
3. r
= 3
in
., � =
7 i
n.
4.
r =
5 i
n.,
� =
11
in.
5. r
= 1
ft,
� =
3 f
t
6. r
= 3
ft,
� =
1.5
ft
7. r
= 1
0 m
m, �
= 2
0 m
m
8. r
= 1
.5 m
m, �
= 4
.5 m
m
9. r
= 6
.2 c
m, �
= 1
.2 c
m
10. r
= 1
0 cm
, � =
15
cm
11. r
= 1
0 m
, � =
2 m
12
. r =
11
m, �
= 1
3 m
Spre
adsh
eet
Act
ivit
yS
urf
ace A
reas o
f C
on
es
Exam
ple
1
Exam
ple
2
A1 2
BC
Sh
eet
1
539.1
m2
94.2
in
2
12.6
ft2 9
42.5
mm
2
144.1
cm
2
377.0
m2
150.8
m2
251.3
in
2
42.4
ft2
28.3
mm
2
785.4
cm
2
829.4
m2
001_
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
NA
ME
DA
TE
PE
RIO
D
Lesson 12-4
Cha
pte
r 12
25
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
Vo
lum
es o
f P
rism
s a
nd
Cylin
ders
Vo
lum
es o
f Pr
ism
s T
he
mea
sure
of
the
amou
nt
of s
pace
th
at a
th
ree-
dim
ensi
onal
fig
ure
en
clos
es i
s th
e vo
lum
e of
th
e fi
gure
. Vol
um
e is
mea
sure
d in
un
its
such
as
cubi
c fe
et, c
ubi
c ya
rds,
or
cubi
c m
eter
s. O
ne
cubi
c u
nit
is
the
volu
me
of a
cu
be
that
mea
sure
s on
e u
nit
on
eac
h e
dge.
Vo
lum
e
of
a P
ris
m
If a
prism
ha
s a
vo
lum
e o
f V
cu
bic
un
its,
a h
eig
ht
of
h u
nits,
an
d e
ach
ba
se
ha
s a
n a
rea
of
B s
qu
are
un
its,
the
n V
= B
h.
F
ind
th
e vo
lum
e of
th
e p
rism
.
7 cm
3 cm4
cm
V =
Bh
V
olu
me o
f a p
rism
=
(7)
(3)(
4)
B =
(7)(
3),
h =
4
=
84
Multip
ly.
Th
e vo
lum
e of
th
e pr
ism
is
84 c
ubi
c ce
nti
met
ers.
F
ind
th
e vo
lum
e of
th
e p
rism
if
the
area
of
each
bas
e is
6.3
sq
uar
e fe
et. 3.
5 ft
base
V =
Bh
V
olu
me o
f a p
rism
=
(6.
3)(3
.5)
B =
6.3
, h
= 3
.5
=
22.
05
Multip
ly.
Th
e vo
lum
e is
22.
05 c
ubi
c fe
et.
Exer
cise
s
Fin
d t
he
volu
me
of e
ach
pri
sm.
1.
8 ft
8 ft
8 ft
2.
3 cm
4 cm
1.5
cm
3.
30°
15 ft12
ft
4.
10 ft
15 ft
12 ft
5.
4 cm
6 cm
2 cm
1.5
cm
6.
7 yd
4 yd
3 yd
12-4
Exam
ple
1Ex
amp
le 2
cubi
c fo
otcu
bic
yard
27 c
ubic
feet
= 1
cubic
yard
512 f
t39 c
m3
467.7
ft3
1800 f
t3
27 c
m3
84 y
d3
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M
Answers (Lesson 12-3 and Lesson 12-4)
A01-A26_GEOCRMC12_890521.indd A11A01-A26_GEOCRMC12_890521.indd A11 6/1/09 12:11:35 PM6/1/09 12:11:35 PM
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pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
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cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A12 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
2
6
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Vo
lum
es o
f P
rism
s a
nd
Cylin
ders
Vo
lum
es o
f C
ylin
der
s T
he
volu
me
of a
cyl
inde
r is
th
e pr
odu
ct o
f th
e
hei
ght
and
the
area
of
the
base
. Wh
en a
sol
id i
s n
ot a
rig
ht
soli
d, u
se
Cav
alie
ri’s
Pri
nic
iple
to
fin
d th
e vo
lum
e. T
he
prin
cipl
e st
ates
th
at i
f tw
o so
lids
hav
e th
e sa
me
hei
ght
and
the
sam
e cr
oss
sect
ion
al a
rea
at e
very
le
vel,
then
th
ey h
ave
the
sam
e vo
lum
e.
F
ind
th
e vo
lum
e of
th
e cy
lin
der
.
4 cm3 cm
V =
πr2 h
V
olu
me o
f a c
ylin
der
=
π(3
)2 (4)
r
= 3
, h
= 4
≈
113
.1
Sim
plif
y.
Th
e vo
lum
e is
abo
ut
113.
1 cu
bic
cen
tim
eter
s.
F
ind
th
e vo
lum
e of
th
e ob
liq
ue
cyli
nd
er.
8 in
.13 in
.
5 in
.
h Use
th
e P
yth
agor
ean
Th
eore
m t
o fi
nd
the
hei
ght
of
the
cyli
nde
r.h
2 +
52
= 1
32 P
yth
agore
an T
heore
m
h
2 =
144
S
implif
y.
h
= 1
2 T
ake t
he s
quare
root
of
each s
ide.
V
= π
r2 h
Volu
me o
f a c
ylin
der
=
π(4
)2 (12
) r
= 4
, h
= 1
2
≈
603
.2
Sim
plif
y.
Th
e V
olu
me
is a
bou
t 60
3.2
cubi
c in
ches
.Ex
erci
ses
Fin
d t
he
volu
me
of e
ach
cyl
ind
er. R
oun
d t
o th
e n
eare
st t
enth
.
1.
2 ft
1 ft
2.
18
cm
2 cm
1
2.6
ft3
226.2
cm
3
3.
12 ft
1.5
ft
4.
20
ft
20 ft
8
4.8
ft3
6
283.2
ft3
5.
10 c
m
13 c
m
6.
1 yd
4 yd
6
52.4
cm
3
12.6
yd
3
12-4
Vo
lum
e o
f
a C
yli
nd
er
If a
cylin
de
r h
as a
vo
lum
e o
f V
cu
bic
un
its,
a h
eig
ht
of
h u
nits,
an
d t
he
ba
se
s h
ave
a r
ad
ius o
f r
un
its,
the
n V
= π
r2h.
Exam
ple
1Ex
amp
le 2
r
h
026_
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OC
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8P
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-4
Cha
pte
r 12
27
Gle
ncoe
Geo
met
ry
12-4
Fin
d t
he
volu
me
of e
ach
pri
sm o
r cy
lin
der
. Rou
nd
to
the
nea
rest
ten
th
if n
eces
sary
.
1.
18 c
m
16 c
m8 cm
2.
6 ft
8 ft
2 ft
2
304 c
m3
96 f
t3
3.
3 m
5 m
13 m
4.
16 in
.22
in.
34 in
.
9
0 m
3
5280 i
n3
5.
15 m
m23
mm
6.
6
yd 10 y
d
1
6,2
57.7
mm
3
226.2
yd
3
Fin
d t
he
volu
me
of e
ach
ob
liq
ue
pri
sm o
r cy
lin
der
. Rou
nd
to
the
nea
rest
te
nth
if
nec
essa
ry.
7.
8.
5 in
.
3 in
.
1224 c
m3
141.4
in
3
Skill
s Pr
acti
ceV
olu
mes o
f P
rism
s a
nd
Cylin
ders
17 c
m
18 c
m
4 cm
026_
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Answers (Lesson 12-4)
A01-A26_GEOCRMC12_890521.indd A12A01-A26_GEOCRMC12_890521.indd A12 6/14/08 9:21:21 PM6/14/08 9:21:21 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A13 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
2
8
Gle
ncoe
Geo
met
ry
Prac
tice
Vo
lum
es o
f P
rism
s a
nd
Cylin
ders
Fin
d t
he
volu
me
of e
ach
pri
sm o
r cy
lin
der
. Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
1.
17 m
10 m
26 m
2.
5 in
.
5 in
.
5 in
.
9 in
.
2
040 m
3
97.4
in
3
3. 16
mm
17.5
mm
4.
7 ft
25 ft
3
518.6
mm
3
923.6
ft3
5.
13 y
d
20 y
d
10 y
d
6. 30
cm
8 cm
2
600 y
d3
6031.9
cm
3
7. A
QU
AR
IUM
Mr.
Gu
tier
rez
purc
has
ed a
cyl
indr
ical
aqu
ariu
m f
or h
is o
ffic
e.
Th
e aq
uar
ium
has
a h
eigh
t of
25
1 −
2 i
nch
es a
nd
a ra
diu
s of
21
inch
es.
a. W
hat
is
the
volu
me
of t
he
aqu
ariu
m i
n c
ubi
c fe
et?
20.4
ft3
b.
If t
her
e ar
e 7.
48 g
allo
ns
in a
cu
bic
foot
, how
man
y ga
llon
s of
wat
er d
oes
the
aqu
ariu
m
hol
d?
152.9
gal
c. I
f a
cubi
c fo
ot o
f w
ater
wei
ghs
abou
t 62
.4 p
oun
ds, w
hat
is
the
wei
ght
of t
he
wat
er i
n
the
aqu
ariu
m t
o th
e n
eare
st f
ive
pou
nds
?
1275 l
b
12-4
026_
044_
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OC
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:10:
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M
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NA
ME
DA
TE
PE
RIO
D
Lesson 12-4
Cha
pte
r 12
29
Gle
ncoe
Geo
met
ry
Wor
d Pr
oble
m P
ract
ice
Vo
lum
es o
f P
rism
s a
nd
Cylin
ders
1. T
RA
SH C
AN
S T
he
Mey
er f
amil
y u
ses
a ki
tch
en t
rash
can
sh
aped
lik
e a
cyli
nde
r.
It h
as a
hei
ght
of
18 i
nch
es a
nd
a ba
se
diam
eter
of
12 i
nch
es.
Wh
at i
s th
e vo
lum
e of
th
e tr
ash
can
? R
oun
d yo
ur
answ
er t
o th
e n
eare
st t
enth
of
a cu
bic
inch
.
2035.8
in
3
2. B
ENC
H I
nsi
de a
lob
by, t
her
e is
a p
iece
of
fu
rnit
ure
for
sit
tin
g. T
he
furn
itu
re i
s sh
aped
lik
e a
sim
ple
bloc
k w
ith
a s
quar
e ba
se 6
fee
t on
eac
h s
ide
and
a h
eigh
t of
1
3 − 5
feet
.
6 ft
6 ft
1ft
3 5
Wh
at i
s th
e vo
lum
e of
th
e se
at?
5
7.6
ft3
3. F
RA
MES
Mar
gare
t m
akes
a s
quar
e fr
ame
out
of f
our
piec
es o
f w
ood.
Eac
h
piec
e of
woo
d is
a
rect
angu
lar
pris
m
wit
h a
len
gth
of
40 c
enti
met
ers,
a
hei
ght
of
4 ce
nti
met
ers,
an
d a
dept
h o
f 6
cen
tim
eter
s.
Wh
at i
s th
e to
tal
volu
me
of t
he
woo
d u
sed
in t
he
fram
e?
3
840 c
m3
4. P
ENC
IL G
RIP
S A
pen
cil
grip
is
shap
ed
like
a t
rian
gula
r pr
ism
wit
h a
cyl
inde
r re
mov
ed f
rom
th
e m
iddl
e. T
he
base
of
the
pris
m i
s a
righ
t is
osce
les
tria
ngl
e w
ith
leg
len
gth
s of
2 c
enti
met
ers.
Th
e di
amet
er o
f th
e ba
se o
f th
e re
mov
ed
cyli
nde
r is
1 c
enti
met
er. T
he
hei
ghts
of
the
pris
m a
nd
the
cyli
nde
r ar
e th
e sa
me,
an
d eq
ual
to
4 ce
nti
met
ers.
Wh
at i
s th
e ex
act
volu
me
of t
he
pen
cil
grip
?
8
- π
cm
3
5. T
UN
NEL
S C
onst
ruct
ion
wor
kers
are
di
ggin
g a
tun
nel
th
rou
gh a
mou
nta
in.
Th
e sp
ace
insi
de t
he
tun
nel
is
goin
g to
be
sh
aped
lik
e a
rect
angu
lar
pris
m. T
he
mou
th o
f th
e tu
nn
el w
ill
be a
rec
tan
gle
20 f
eet
hig
h a
nd
50 f
eet
wid
e an
d th
e le
ngt
h o
f th
e tu
nn
el w
ill
be 9
00 f
eet.
a. W
hat
will
the
vol
ume
of t
he t
unne
l be?
900,0
00 f
t3
b.
If i
nst
ead
of a
rec
tan
gula
r sh
ape,
th
e tu
nn
el h
ad a
sem
icir
cula
r sh
ape
wit
h
a 50
-foo
t di
amet
er, w
hat
wou
ld b
e it
s vo
lum
e? R
oun
d yo
ur
answ
er t
o th
e n
eare
st c
ubi
c fo
ot.
883,5
73 f
t3
18 in
.
12 in
.
12-4
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Answers (Lesson 12-4)
A01-A26_GEOCRMC12_890521.indd A13A01-A26_GEOCRMC12_890521.indd A13 6/14/08 9:21:24 PM6/14/08 9:21:24 PM
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pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A14 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
3
0
Gle
ncoe
Geo
met
ry
Vis
ible
Su
rface A
rea
Use
pap
er, s
ciss
ors,
an
d t
ape
to m
ake
five
cu
bes
th
at h
ave
one-
inch
ed
ges.
A
rran
ge t
he
cub
es t
o fo
rm e
ach
sh
ape
show
n. T
hen
fin
d t
he
volu
me
and
th
e vi
sib
le s
urf
ace
area
. In
oth
er w
ord
s, d
o n
ot i
ncl
ud
e th
e ar
ea o
f su
rfac
e co
vere
d b
y ot
her
cu
bes
or
by
the
tab
le o
r d
esk
.
1.
2.
v
olu
me
= 4 i
n3
vol
um
e =
4 i
n3
v
isib
le s
urf
ace
area
= 14 i
n2
vis
ible
su
rfac
e ar
ea =
15 i
n2
3.
4.
5.
v
olu
me
= 5 i
n3
vol
um
e =
5 i
n3
vol
um
e =
5 i
n3
v
isib
le s
urfa
ce a
rea
= 1
7 in
2
vis
ible
sur
face
are
a =
19 in
2
vis
ible
sur
face
are
a =
19 in
2
6. F
ind
the
volu
me
and
the
visi
ble
surf
ace
ar
ea o
f th
e fi
gure
at
the
righ
t.
v
olu
me
= 136 i
n3
v
isib
le s
urf
ace
area
= 164 i
n2
4 in
.
4 in
.
3 in
.
3 in
.
3 in
.
8 in
.
3 in
. 5 in
.
5 in
.
12-4
Enri
chm
ent
026_
044_
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OC
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NA
ME
DA
TE
PE
RIO
D
Lesson 12-5
Cha
pte
r 12
31
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
Vo
lum
es o
f P
yra
mid
s a
nd
Co
nes
Vo
lum
es o
f Py
ram
ids
Th
is f
igu
re s
how
s a
pris
m a
nd
a py
ram
id
that
hav
e th
e sa
me
base
an
d th
e sa
me
hei
ght.
It
is c
lear
th
at t
he
volu
me
of t
he
pyra
mid
is
less
th
an t
he
volu
me
of t
he
pris
m. M
ore
spec
ific
ally
, th
e vo
lum
e of
th
e py
ram
id i
s on
e-th
ird
of t
he
volu
me
of t
he
pris
m.
F
ind
th
e vo
lum
e of
th
e sq
uar
e p
yram
id.
V =
1 −
3 B
h
Volu
me o
f a p
yra
mid
=
1 −
3 (8
)(8)
10
B =
(8)(
8),
h =
10
≈
213
.3
Multip
ly.
Th
e vo
lum
e is
abo
ut
213.
3 cu
bic
feet
.
Exer
cise
sF
ind
th
e vo
lum
e of
eac
h p
yram
id. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
1.
12 ft
8 ft
10 ft
2.
10 ft
6 ft
15 ft
3
20 f
t3
120 f
t3
3.
4 cm
8 cm
12 c
m
4.
18 ft
regu
lar
hexa
gon
6 ft
1
10.9
cm
3
561.2
ft3
5.
15 in
.
15 in
.
16 in
.
6.
6 yd
8 yd
5 yd
1
200 i
n3
64 y
d3
8 ft
8 ft
10 ft
12-5
Vo
lum
e o
f
a P
yra
mid
If a
pyra
mid
ha
s a
vo
lum
e o
f V
cu
bic
un
its,
a h
eig
ht
of
h u
nits,
an
d a
ba
se
with
an
are
a o
f B
sq
ua
re u
nits,
the
n V
= 1
−
3 B
h.
Exam
ple
026_
044_
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OC
RM
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521.
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10/0
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M
Answers (Lesson 12-4 and Lesson 12-5)
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An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A15 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
3
2
Gle
ncoe
Geo
met
ry
Vo
lum
es o
f C
on
es F
or a
con
e, t
he
volu
me
is o
ne-
thir
d th
e pr
odu
ct o
f th
e
hei
ght
and
the
area
of
the
base
. Th
e ba
se o
f a
con
e is
a c
ircl
e, s
o th
e ar
ea o
f th
e ba
se i
s π
r2 .
F
ind
th
e vo
lum
e of
th
e co
ne.
V
= 1 −
3 π
r2 h
Volu
me o
f a c
one
=
1 −
3 π
(5)2 1
2 r
= 5
, h
= 1
2
≈
314
.2
Sim
plif
y.
Th
e vo
lum
e of
th
e co
ne
is a
bou
t 31
4.2
cubi
c ce
nti
met
ers.
Exer
cise
sF
ind
th
e vo
lum
e of
eac
h c
one.
Rou
nd
to
the
nea
rest
ten
th.
1.
6 cm
10 c
m
2.
8 ft
10 ft
3
01.6
cm
3
670.2
ft3
3.
30 in
.
12 in
.
4.
45
°18
yd 20
yd
1
131.0
in
3
1332.9
yd
3
5.
26 ft
20 ft
6.
16 c
m
45°
2
513.3
ft3
3
79.1
cm
3
12 c
m
5 cm
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Vo
lum
es o
f P
yra
mid
s a
nd
Co
nes
12-5 V
olu
me
of
a C
on
e
If a
co
ne
ha
s a
vo
lum
e o
f V
cu
bic
un
its,
a h
eig
ht
of
h u
nits,
an
d t
he
ba
se
s h
ave
a r
ad
ius o
f r
un
its,
the
n V
= 1
−
3 π
r2 h.
Exam
ple
r
h
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-5
Cha
pte
r 12
3
3
Gle
ncoe
Geo
met
ry
12-5
Skill
s Pr
acti
ceV
olu
mes o
f P
yra
mid
s a
nd
Co
nes
Fin
d t
he
volu
me
of e
ach
pyr
amid
or
con
e. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
1.
2.
6
6.7
ft3
7
4.7
cm
3
3.
4.
3
57.8
in
3
3769.9
m3
5.
6.
1
231.5
yd
3
1210.6
mm
3
Fin
d t
he
volu
me
of e
ach
ob
liq
ue
pyr
amid
or
con
e. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
7.
8.
31 f
t3
452.4
cm
3
5 ft
5 ft
8 ft
4 cm
7 cm
8 cm
8 in
.10
in.
14 in
.25
m12 m
25 y
d
14 y
d66
°
18 m
m
4 ft
4 ft
6 ft
12 c
m6 cm
026_
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Answers (Lesson 12-5)
A01-A26_GEOCRMC12_890521.indd A15A01-A26_GEOCRMC12_890521.indd A15 6/14/08 9:21:31 PM6/14/08 9:21:31 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A16 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
3
4
Gle
ncoe
Geo
met
ry
Prac
tice
Vo
lum
es o
f P
yra
mid
s a
nd
Co
nes
Fin
d t
he
volu
me
of e
ach
pyr
amid
or
con
e. R
oun
d t
o th
e n
eare
st t
enth
if
nec
essa
ry.
1.
9.2
yd9.
2 yd
13 y
d
2.
12.5
cm
25 c
m
23 c
m
3
17.5
yd
3
2395.8
cm
3
3.
19 ft
9 ft
4.
52°
12 m
m
1
419.4
ft3
1
104.6
mm
3
5.
6 in
.6
in.
11 in
. 6.
37 ft
11 ft
1
32 i
n3
4688.3
ft3
7. C
ON
STR
UC
TIO
N M
r. G
anty
bu
ilt
a co
nic
al s
tora
ge s
hed
. Th
e ba
se o
f th
e sh
ed i
s 4
met
ers
in d
iam
eter
an
d th
e h
eigh
t of
th
e sh
ed i
s 3.
8 m
eter
s. W
hat
is
the
volu
me
of
the
shed
?
a
bo
ut
15.9
m3
8. H
ISTO
RY
Th
e st
art
of t
he
pyra
mid
age
beg
an w
ith
Kin
g Z
oser
’s p
yram
id, e
rect
ed i
n t
he
27th
cen
tury
B.C
. In
its
ori
gin
al s
tate
, it
stoo
d 62
met
ers
hig
h w
ith
a r
ecta
ngu
lar
base
th
at m
easu
red
140
met
ers
by 1
18 m
eter
s. F
ind
the
volu
me
of t
he
orig
inal
pyr
amid
.
a
bo
ut
341,4
13.3
m3
12-5
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0P
M
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-5
Cha
pte
r 12
35
Gle
ncoe
Geo
met
ry
1. I
CE
CR
EAM
DIS
HES
Th
e pa
rt o
f a
dish
de
sign
ed f
or i
ce c
ream
is
shap
ed l
ike
an
ups
ide-
dow
n c
one.
Th
e ba
se o
f th
e co
ne
has
a r
adiu
s of
2 i
nch
es a
nd
the
hei
ght
is 1
.2 i
nch
es.
Wh
at i
s th
e vo
lum
e of
th
e co
ne?
Rou
nd
you
r an
swer
to
the
nea
rest
hu
ndr
edth
.
5
.03 i
n3
2. G
REE
NH
OU
SES
A g
reen
hou
se h
as t
he
shap
e of
a s
quar
e py
ram
id. T
he
base
has
a
side
len
gth
of
30 y
ards
. Th
e h
eigh
t of
th
e gr
een
hou
se i
s 18
yar
ds.
18 yd
30
yd
Wh
at i
s th
e vo
lum
e of
th
e gr
een
hou
se?
5
400 y
d3
3. T
EEPE
E C
aitl
yn m
ade
a te
epee
for
a
clas
s pr
ojec
t. H
er t
eepe
e h
ad a
dia
met
er
of 6
fee
t. T
he
angl
e th
e si
de o
f th
e te
epee
m
ade
wit
h t
he
grou
nd
was
65°
.
65˚
Wh
at w
as t
he
volu
me
of t
he
teep
ee?
Rou
nd
you
r an
swer
to
the
nea
rest
h
un
dred
th.
6
0.6
3 f
t3
4. S
CU
LPTI
NG
A s
culp
tor
wan
ts t
o re
mov
e st
one
from
a c
ylin
dric
al b
lock
3 f
eet
hig
h
and
turn
it
into
a c
one.
Th
e di
amet
er o
f th
e ba
se o
f th
e co
ne
and
cyli
nde
r is
2
feet
.
Wh
at i
s th
e vo
lum
e of
th
e st
one
that
th
e sc
ulp
tor
mu
st r
emov
e? R
oun
d yo
ur
answ
er t
o th
e n
eare
st h
un
dred
th.
6
.28 f
t3
5. S
TAG
ESA
sta
ge h
as t
he
form
of
a sq
uar
e py
ram
id w
ith
th
e to
p sl
iced
off
al
ong
a pl
ane
para
llel
to
the
base
. Th
e si
de l
engt
h o
f th
e to
p sq
uar
e is
12
feet
an
d th
e si
de l
engt
h o
f th
e bo
ttom
squ
are
is 1
6 fe
et. T
he
hei
ght
of t
he
stag
e is
3
feet
.
12
feet
16
feet
3 fe
et
a. W
hat
is
the
volu
me
of t
he
enti
re
squ
are
pyra
mid
th
at t
he
stag
e is
pa
rt o
f?
1024 f
t3
b.
Wh
at i
s th
e vo
lum
e of
th
e to
p of
th
e py
ram
id t
hat
is
rem
oved
to
get
the
stag
e?
432 f
t3
c. W
hat
is
the
volu
me
of t
he
stag
e?
592 f
t3
Wor
d Pr
oble
m P
ract
ice
Vo
lum
es o
f P
yra
mid
s a
nd
Co
nes
12-5
026_
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Answers (Lesson 12-5)
A01-A26_GEOCRMC12_890521.indd A16A01-A26_GEOCRMC12_890521.indd A16 6/1/09 12:12:07 PM6/1/09 12:12:07 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A17 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
3
6
Gle
ncoe
Geo
met
ry
Enri
chm
ent
Fru
stu
ms
A f
rust
um
is
a fi
gure
for
med
wh
en a
pla
ne
inte
rsec
ts a
pyr
amid
or
con
e so
th
at t
he
plan
e is
par
alle
l to
th
e so
lid’
s ba
se. T
he
fru
stu
m i
s th
e pa
rt o
f th
e so
lid
betw
een
th
e pl
ane
and
the
base
. To
fin
d th
e vo
lum
e of
a f
rust
um
, th
e ar
eas
of b
oth
bas
es m
ust
be
calc
ula
ted
and
use
d in
th
e fo
rmu
la.
V =
1 −
3 h
(B1
+ B
2 +
√ �
�
B1B
2 ),w
her
e h
= h
eigh
t (p
erpe
ndi
cula
r di
stan
ce b
etw
een
th
e ba
ses)
,B
1 =
are
a of
top
bas
e, a
nd
B2
= a
rea
of b
otto
m b
ase.
Des
crib
e th
e sh
ape
of t
he
bas
es o
f ea
ch f
rust
um
. Th
en f
ind
th
e vo
lum
e. R
oun
d t
o th
e n
eare
st t
enth
.
1.
13 c
m 6 cm
9 cm
5 cm
19.5
cm
2.
7.5
in.
4.5
in.
3 in
.
r
ecta
ng
les;
617.5
cm
3
cir
cle
s;
335.8
in
3
3.
8 m
6 m
12 m
4.5
m2.
25 m3
m
5 m
4.
12 ft
13 ft
7 ft
tra
pezo
ids;
151.6
m3
cir
cle
s;
3480.9
ft3
12-5
026_
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-6
Cha
pte
r 12
37
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
Su
rface A
reas a
nd
Vo
lum
es o
f S
ph
ere
s
Surf
ace
Are
as o
f Sp
her
es Y
ou c
an t
hin
k of
th
e su
rfac
e ar
ea o
f a
sph
ere
as
th
e to
tal
area
of
all
of t
he
non
over
lapp
ing
stri
ps i
t w
ould
tak
e to
cov
er
the
sph
ere.
If
r is
th
e ra
diu
s of
th
e sp
her
e, t
hen
th
e ar
ea o
f a
grea
t ci
rcle
of
the
sph
ere
is π
r2 . T
he
tota
l su
rfac
e ar
ea o
f th
e sp
her
e is
fou
r ti
mes
th
e ar
ea
of a
gre
at c
ircl
e.
F
ind
th
e su
rfac
e ar
ea o
f a
sph
ere
to t
he
nea
rest
ten
th
if t
he
rad
ius
of t
he
sph
ere
is 6
cen
tim
eter
s.
S =
4π
r2 S
urf
ace a
rea o
f a s
phere
=
4π
(6)2
r =
6
≈
452
.4
Sim
plif
y.
Th
e su
rfac
e ar
ea i
s 45
2.4
squ
are
cen
tim
eter
s.
Exer
cise
sF
ind
th
e su
rfac
e ar
ea o
f ea
ch s
ph
ere
or h
emis
ph
ere.
Rou
nd
to
the
nea
rest
ten
th.
1.
5 m
314.2
m2
2.
7 in
153.9
in
2
3.
3 ft
84.8
ft2
4.
9 cm
190.9
cm
2
5. s
pher
e: c
ircu
mfe
ren
ce o
f gr
eat
circ
le =
π c
m 3.1
cm
2
6.
h
em
isp
he
re
: a
re
a o
f g
re
at c
irc
le
≈ 4
π f
t2 37.7
ft2
r
6 cm
Su
rfa
ce
Are
a
of
a S
ph
ere
If a
sp
he
re h
as a
su
rfa
ce
are
a o
f S
sq
ua
re u
nits a
nd
a r
ad
ius o
f r
un
its,
the
n S
= 4
πr2
.
Exam
ple
12-6
026_
044_
GE
OC
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1P
M
Answers (Lesson 12-5 and Lesson 12-6)
A01-A26_GEOCRMC12_890521.indd A17A01-A26_GEOCRMC12_890521.indd A17 6/1/09 12:12:13 PM6/1/09 12:12:13 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A18 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
3
8
Gle
ncoe
Geo
met
ry
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Su
rface A
reas a
nd
Vo
lum
es o
f S
ph
ere
s
12-6
Vo
lum
es o
f Sp
her
es A
sph
ere
has
on
e ba
sic
mea
sure
men
t, t
he
le
ngt
h o
f it
s ra
diu
s. I
f yo
u k
now
th
e le
ngt
h o
f th
e ra
diu
s of
a s
pher
e, y
ou
can
cal
cula
te i
ts v
olu
me.
F
ind
th
e vo
lum
e of
a s
ph
ere
wit
h r
adiu
s 8
cen
tim
eter
s.
V =
4 −
3 π
r3 V
olu
me o
f a s
phere
=
4 −
3 π
(8)3
r =
8
≈
214
4.7
Sim
plif
y.
Th
e vo
lum
e is
abo
ut
2144
.7 c
ubi
c ce
nti
met
ers.
Exer
cise
sF
ind
th
e vo
lum
e of
eac
h s
ph
ere
or h
emis
ph
ere.
Rou
nd
to
the
nea
rest
ten
th.
1.
5 ft
2.
6
in.
3.
16 in
.
5
23.6
ft3
4
52.4
in
3
8578.6
in
3
4. h
emis
pher
e: r
adiu
s 5
in.
261.8
in
3
5. s
pher
e: c
ircu
mfe
ren
ce o
f gr
eat
circ
le ≈
25
ft 2
63.9
ft3
6. h
emis
pher
e: a
rea
of g
reat
cir
cle
≈ 5
0 m
2 133.0
m3
r
8 cm
Vo
lum
e o
f
a S
ph
ere
If a
sp
he
re h
as a
vo
lum
e o
f V
cu
bic
un
its a
nd
a r
ad
ius o
f r
un
its,
the
n V
= 4
−
3 π
r3.
Exam
ple
026_
044_
GE
OC
RM
C12
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521.
indd
383/
27/1
010
:47:
18P
M
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-6
Cha
pte
r 12
39
Gle
ncoe
Geo
met
ry
Skill
s Pr
acti
ceS
urf
ace A
reas a
nd
Vo
lum
es o
f S
ph
ere
sF
ind
th
e su
rfac
e ar
ea o
f ea
ch s
ph
ere
or h
emis
ph
ere.
Rou
nd
to
the
nea
rest
ten
th.
1.
7 in
.
2.
32 m
6
15.8
in
2
3217.0
m2
3. h
emis
pher
e: r
adiu
s of
gre
at c
ircl
e =
8 y
d 603.2
yd
2
4. s
pher
e: a
rea
of g
reat
cir
cle
≈ 2
8.6
in2
114.4
in
2
Fin
d t
he
volu
me
of e
ach
sp
her
e or
hem
isp
her
e. R
oun
d t
o th
e n
eare
st t
enth
.
5.
16.2
cm
6.
94.8
ft
2
226.1
cm
3
446,0
91.2
ft3
7. h
emis
pher
e: d
iam
eter
= 4
8 yd
28,9
52.9
yd
3
8. s
pher
e: c
ircu
mfe
ren
ce o
f a
grea
t ci
rcle
≈ 2
6 m
296.8
m3
9. s
pher
e: d
iam
eter
= 1
0 in
. 523.6
in
3
12-6
026_
044_
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OC
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C12
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395/
30/0
93:
09:1
6P
M
Answers (Lesson 12-6)
A01-A26_GEOCRMC12_890521.indd A18A01-A26_GEOCRMC12_890521.indd A18 3/27/10 10:56:38 PM3/27/10 10:56:38 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A19 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
4
0
Gle
ncoe
Geo
met
ry
12-6
Prac
tice
Su
rface A
reas a
nd
Vo
lum
es o
f S
ph
ere
sF
ind
th
e su
rfac
e ar
ea o
f ea
ch s
ph
ere
or h
emis
ph
ere.
Rou
nd
to
the
nea
rest
ten
th.
1.
6.5
cm
2.
89 ft
5
30.9
cm
2
24,8
84.6
ft2
3. h
emis
pher
e: r
adiu
s of
gre
at c
ircl
e =
8.4
in
. 665.0
in
2
4. s
pher
e: a
rea
of g
reat
cir
cle
≈ 2
9.8
m2
119.2
m2
Fin
d t
he
volu
me
of e
ach
sp
her
e or
hem
isp
her
e. R
oun
d t
o th
e n
eare
st t
enth
.
5.
12.3
2 ft
6.
32
m
7
832.9
ft3
8578.6
m3
7. h
emis
pher
e: d
iam
eter
= 1
8 m
m 1
526.8
mm
3
8. s
pher
e: c
ircu
mfe
ren
ce ≈
36
yd 7
87.9
yd
3
9. s
pher
e: r
adiu
s =
12.
4 in
. 7986.4
in
3
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NA
ME
DA
TE
PE
RIO
D
Lesson 12-6
Cha
pte
r 12
41
Gle
ncoe
Geo
met
ry
Wor
d Pr
oble
m P
ract
ice
Su
rface A
reas a
nd
Vo
lum
es o
f S
ph
ere
s 1
. OR
AN
GES
Man
dy c
uts
a s
pher
ical
or
ange
in
hal
f al
ong
a gr
eat
circ
le. I
f th
e ra
diu
s of
th
e or
ange
is
2 in
ches
, wh
at i
s th
e ar
ea o
f th
e cr
oss
sect
ion
th
at M
andy
cu
t? R
oun
d yo
ur
answ
er t
o th
e n
eare
st
hu
ndr
edth
.
1
2.5
7 i
n2
2. B
ILLI
AR
DS
A b
illi
ard
ball
set
con
sist
s of
16
sph
eres
, eac
h 2
1 − 4 i
nch
es i
n
diam
eter
. Wh
at i
s th
e to
tal
volu
me
of
a co
mpl
ete
set
of b
illi
ard
ball
s? R
oun
d yo
ur
answ
er t
o th
e n
eare
st t
hou
san
dth
of
a c
ubi
c in
ch.
95.4
26 i
n3
3. M
OO
NS
OF
SATU
RN
Th
e pl
anet
S
atu
rn h
as s
ever
al m
oon
s. T
hes
e ca
n
be m
odel
ed a
ccu
rate
ly b
y sp
her
es.
Sat
urn
’s l
arge
st m
oon
Tit
an h
as a
ra
diu
s of
abo
ut
2575
kil
omet
ers.
Wh
at i
s th
e ap
prox
imat
e su
rfac
e ar
ea o
f T
itan
? R
oun
d yo
ur
answ
er t
o th
e n
eare
st t
enth
.
83,3
22,8
91.2
km
2
4. T
HE
ATM
OSP
HER
EA
bou
t 99
% o
f
Ear
th’s
atm
osph
ere
is c
onta
ined
in
a
31-k
ilom
eter
th
ick
laye
r th
at e
nw
raps
th
e pl
anet
. Th
e E
arth
its
elf
is a
lmos
t a
sph
ere
wit
h r
adiu
s 63
78 k
ilom
eter
s.
Wh
at i
s th
e ra
tio
of t
he
volu
me
of t
he
atm
osph
ere
to t
he
volu
me
of E
arth
? R
oun
d yo
ur
answ
er t
o th
e n
eare
st
thou
san
dth
.
0.0
15
5. C
UB
ESM
arcu
s bu
ilds
a s
pher
e in
side
of
a cu
be. T
he
sph
ere
fits
sn
ugl
y in
side
th
e cu
be s
o th
at t
he
sph
ere
tou
ches
th
e cu
be
at o
ne
poin
t on
eac
h s
ide.
Th
e si
de
len
gth
of
the
cube
is
2 in
ches
.
a. W
hat
is
the
surf
ace
area
of
the
cube
?
24 i
n2
b.
Wh
at i
s th
e su
rfac
e ar
ea o
f th
e sp
her
e? R
oun
d yo
ur
answ
ers
to t
he
nea
rest
hu
ndr
edth
.
12.5
7 i
n2
c. W
hat
is
the
rati
o of
th
e su
rfac
e ar
ea
of t
he
cube
to
the
surf
ace
area
of
the
sph
ere?
Rou
nd
you
r an
swer
to
the
nea
rest
hu
ndr
edth
.
1.9
1
12-6
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Answers (Lesson 12-6)
A01-A26_GEOCRMC12_890521.indd A19A01-A26_GEOCRMC12_890521.indd A19 6/1/09 12:12:25 PM6/1/09 12:12:25 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A20 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
4
2
Gle
ncoe
Geo
met
ry
Enri
chm
ent
Sp
here
s a
nd
Den
sit
yT
he
den
sity
of
a m
etal
is
a ra
tio
of i
ts m
ass
to i
ts v
olu
me.
For
ex
ampl
e, t
he
mas
s of
alu
min
um
is
2.7
gram
s pe
r cu
bic
cen
tim
eter
. H
ere
is a
lis
t of
sev
eral
met
als
and
thei
r de
nsi
ties
.
Alu
min
um
2.
7 g/
cm3
Cop
per
8.96
g/c
m3
Gol
d 19
.32
g/cm
3 Ir
on
7.87
4 g/
cm3
Lea
d 11
.35
g/cm
3 P
lati
nu
m
21.4
5 g/
cm3
Sil
ver
10.5
0 g/
cm3
To
calc
ula
te t
he
mas
s of
a p
iece
of
met
al, m
ult
iply
vol
um
e by
den
sity
.
F
ind
th
e m
ass
of a
sil
ver
bal
l th
at i
s 0.
8 cm
in
dia
met
er.
M =
D ·
V
= 1
0.5
· 4 −
3 π
(0.4
)3
≈ 1
0.5(
0.27
)≈
2.8
1
Th
e m
ass
is a
bou
t 2.
81 g
ram
s.
Exer
cise
sF
ind
th
e m
ass
of e
ach
met
al b
all
des
crib
ed. A
ssu
me
the
bal
ls a
re
sph
eric
al. R
oun
d y
our
answ
ers
to t
he
nea
rest
ten
th.
1. a
cop
per
ball
1.2
cm
in
dia
met
er 8
.1 g
2. a
gol
d ba
ll 0
.6 c
m i
n d
iam
eter
2.2
g
3. a
n a
lum
inu
m b
all
wit
h r
adiu
s 3
cm 3
05.4
g
4. a
pla
tin
um
bal
l w
ith
rad
ius
0.7
cm 3
0.8
g
Sol
ve. A
ssu
me
the
bal
ls a
re s
ph
eric
al. R
oun
d y
our
answ
ers
to t
he
nea
rest
ten
th.
5. A
lea
d ba
ll w
eigh
s 32
6 gr
ams.
Fin
d th
e ra
diu
s of
th
e ba
ll t
o th
e n
eare
st
ten
th o
f a
cen
tim
eter
. 1.9
cm
6. A
n i
ron
bal
l w
eigh
s 80
4 gr
ams.
Fin
d th
e di
amet
er o
f th
e ba
ll t
o th
e n
eare
st t
enth
of
a ce
nti
met
er.
5.8
cm
7. A
sil
ver
ball
an
d a
copp
er b
all
each
hav
e a
diam
eter
of
3.5
cen
tim
eter
s.
Wh
ich
wei
ghs
mor
e? H
ow m
uch
mor
e? s
ilver;
34.6
g
8. A
n a
lum
inu
m b
all
and
a le
ad b
all
each
hav
e a
radi
us
of 1
.2 c
enti
met
ers.
W
hic
h w
eigh
s m
ore?
How
mu
ch m
ore?
lead
; 62.6
g
12-6
Exam
ple
026_
044_
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OC
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521.
indd
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09:3
1P
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-7
Cha
pte
r 12
43
Gle
ncoe
Geo
met
ry
12-7
Stud
y G
uide
and
Inte
rven
tion
Sp
heri
cal
Geo
metr
y
Geo
met
ry O
n A
Sp
her
e U
p to
now
, we
hav
e be
en s
tudy
ing
Eu
clid
ean
geo
met
ry,
wh
ere
a pl
ane
is a
fla
t su
rfac
e m
ade
up
of p
oin
ts t
hat
ext
ends
in
fin
itel
y in
all
dir
ecti
ons.
In
sp
her
ical
geo
met
ry, a
pla
ne
is t
he
surf
ace
of a
sph
ere.
Nam
e ea
ch o
f th
e fo
llow
ing
on s
ph
ere
K.
a. t
wo
lin
es c
onta
inin
g th
e p
oin
t F
� ��
EG
an
d � �
��
BH
are
lin
es o
n s
pher
e K
th
at c
onta
in t
he
poin
t F
b. a
lin
e se
gmen
t co
nta
inin
g th
e p
oin
t J
−
−
ID
is
a se
gmen
t on
sph
ere
K t
hat
con
tain
s th
e po
int
J
c. a
tri
angl
e
�
AH
I is
a t
rian
gle
on s
pher
e K
Exer
cise
sN
ame
two
lin
es c
onta
inin
g p
oin
t Z
, a s
egm
ent
con
tain
ing
poi
nt
R, a
nd
a t
rian
gle
in e
ach
of
the
foll
owin
g sp
her
es.
1.
F
2.
M
�
⎯
�
WT a
nd
�
⎯
�
SV
, −
−
X
U a
nd
�S
RT
�
⎯
�
AD
an
d �
⎯
�
EB
, −
−
G
C a
nd
�G
RZ
Det
erm
ine
wh
eth
er f
igu
re u
on
eac
h o
f th
e sp
her
es s
how
n i
s a
lin
e in
sp
her
ical
ge
omet
ry.
3.
No
4.
N
o
5. G
EOG
RA
PHY
Lin
es o
f la
titu
de r
un
hor
izon
tall
y ac
ross
th
e su
rfac
e of
Ear
th. A
re t
her
e an
y li
nes
of
lati
tude
th
at a
re g
reat
cir
cles
? E
xpla
in.
Y
es,
the e
qu
ato
r is
a g
reat
cir
cle
. N
o o
ther
lin
e o
f la
titu
de i
s a
gre
at
cir
cle
.
Exam
ple
K
026_
044_
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OC
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09:3
6P
M
Answers (Lesson 12-6 and Lesson 12-7)
A01-A26_GEOCRMC12_890521.indd A20A01-A26_GEOCRMC12_890521.indd A20 6/1/09 12:12:32 PM6/1/09 12:12:32 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A21 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
4
4
Gle
ncoe
Geo
met
ry
Co
mp
arin
g E
ucl
idea
n a
nd
Sp
her
ical
Geo
met
ries
Som
e po
stu
late
s an
d pr
oper
ties
of
Eu
clid
ean
geo
met
ry a
re t
rue
in s
pher
ical
geo
met
ry. O
ther
s ar
e n
ot t
rue
or a
re
tru
e on
ly u
nde
r ce
rtai
n c
ircu
mst
ance
s.
Tel
l w
het
her
th
e fo
llow
ing
pos
tula
te o
r p
rop
erty
of
pla
ne
Eu
clid
ean
ge
omet
ry h
as a
cor
resp
ond
ing
stat
emen
t in
sp
her
ical
geo
met
ry. I
f so
, wri
te t
he
corr
esp
ond
ing
stat
emen
t. I
f n
ot, e
xpla
in y
our
reas
onin
g.
Giv
en a
ny
lin
e, t
her
e ar
e an
in
fin
ite
nu
mb
er o
f p
aral
lel
lin
es.
On
th
e sp
her
e to
th
e ri
ght,
if
we
are
give
n l
ine
m w
e se
e th
at i
t go
es t
hro
ugh
th
e po
les
of t
he
sph
ere.
If
we
try
to m
ake
any
oth
er
lin
e on
th
e sp
her
e, i
t w
ould
in
ters
ect
lin
e m
at
exac
tly
2 po
ints
. T
his
pro
pert
y is
not
tru
e in
sph
eric
al g
eom
etry
.A
cor
resp
ondi
ng
stat
emen
t in
sph
eric
al g
eom
etry
wou
ld b
e:
“Giv
en a
ny
lin
e, t
her
e ar
e n
o pa
rall
el l
ines
.”
Exer
cise
sT
ell
wh
eth
er t
he
foll
owin
g p
ostu
late
or
pro
per
ty o
f p
lan
e E
ucl
idea
n
geom
etry
has
a c
orre
spon
din
g st
atem
ent
in s
ph
eric
al g
eom
etry
. If
so,
wri
te t
he
corr
esp
ond
ing
stat
emen
t. I
f n
ot, e
xpla
in y
our
reas
onin
g.
1. I
f tw
o n
onid
enti
cal
lin
es i
nte
rsec
t at
a p
oin
t, t
hey
do
not
in
ters
ect
agai
n.
N
o.
If t
wo
no
nid
en
tical
lin
es i
nte
rsect
at
a p
oin
t, t
hey i
nte
rsect
ag
ain
o
n t
he o
pp
osit
e s
ide o
f th
e s
ph
ere
.
2. G
iven
a l
ine
and
a po
int
on t
he
lin
e, t
her
e is
on
ly o
ne
perp
endi
cula
r li
ne
goin
g th
rou
gh
that
poi
nt.
Y
es.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
3. G
iven
tw
o pa
rall
el l
ines
an
d a
tran
sver
sal,
alte
rnat
e in
teri
or a
ngl
es a
re c
ongr
uen
t.
No
. T
here
are
no
para
llel
lin
es i
n s
ph
eri
cal
geo
metr
y.
4. I
f tw
o li
nes
are
per
pen
dicu
lar
to a
th
ird
lin
e, t
hey
are
par
alle
l.
No
. T
here
are
no
para
llel
lin
es i
n s
ph
eri
cal
geo
metr
y.
5. T
hre
e n
onco
llin
ear
poin
ts d
eter
min
e a
tria
ngl
e.
Yes.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
6. A
lar
gest
an
gle
of a
tri
angl
e is
opp
osit
e th
e la
rges
t si
de.
Y
es.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
12-7
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Sp
heri
cal
Geo
metr
y
Exam
ple
026_
044_
GE
OC
RM
C12
_890
521.
indd
444/
10/0
89:
12:3
3P
M
Lesson 12-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
45
Gle
ncoe
Geo
met
ry
Nam
e tw
o li
nes
con
tain
ing
poi
nt
K, a
seg
men
t co
nta
inin
g p
oin
t T
, an
d a
tri
angl
e in
eac
h o
f th
e fo
llow
ing
sph
eres
.
1.
C
2.
L
Det
erm
ine
wh
eth
er f
igu
re u
on
eac
h o
f th
e sp
her
es s
how
n i
s a
lin
e in
sp
her
ical
geo
met
ry.
3.
4.
bask
etba
ll
Tel
l w
het
her
th
e fo
llow
ing
pos
tula
te o
r p
rop
erty
of
pla
ne
Eu
clid
ean
geo
met
ry h
as
a co
rres
pon
din
g st
atem
ent
in s
ph
eric
al g
eom
etry
. If
so, w
rite
th
e co
rres
pon
din
g st
atem
ent.
If
not
, exp
lain
you
r re
ason
ing.
5. I
f tw
o li
nes
for
m v
erti
cal
angl
es, t
hen
th
e an
gles
are
equ
al i
n m
easu
re.
6. I
f tw
o li
nes
mee
t a
thir
d li
ne
at t
he
sam
e an
gle,
th
ose
lin
es a
re p
aral
lel.
7. T
wo
lin
es m
eet
at t
wo
90°
angl
es o
r th
ey m
eet
at a
ngl
es w
hos
e su
m i
s 18
0°.
8. T
hre
e n
on-p
aral
lel
lin
es d
ivid
e th
e pl
ane
into
7 s
epar
ate
part
s.
Skill
s Pr
acti
ceS
ph
eri
cal
Geo
metr
y
12-7
�
⎯
�
SF a
nd
�
⎯
�
IH , −
−
S
G ,
an
d �
STH
�
⎯
�
BD
an
d �
⎯
�
ET ,
−−
C
A ,
an
d �
ATE
N
o
Yes
Y
es.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
N
o.
Th
ere
are
no
para
llel
lin
es i
n s
ph
eri
cal
geo
metr
y.
Y
es.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
N
o.
Th
ree l
ines d
ivid
e t
he p
lan
e i
nto
6 o
r 7 s
ep
ara
te p
art
s.
045_
055_
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:36
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Answers (Lesson 12-7)
A01-A26_GEOCRMC12_890521.indd A21A01-A26_GEOCRMC12_890521.indd A21 6/2/09 5:32:13 PM6/2/09 5:32:13 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A22 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
4
6
Gle
ncoe
Geo
met
ry
Nam
e tw
o li
nes
con
tain
ing
poi
nt
K, a
seg
men
t co
nta
inin
g p
oin
t T
, an
d a
tri
angl
e in
eac
h o
f th
e fo
llow
ing
sph
eres
.
1.
L
2.
M
Det
erm
ine
wh
eth
er f
igu
re u
on
eac
h o
f th
e sp
her
es s
how
n i
s a
lin
e in
sp
her
ical
ge
omet
ry.
3.
tenn
is b
all
4.
Tel
l w
het
her
th
e fo
llow
ing
pos
tula
te o
r p
rop
erty
of
pla
ne
Eu
clid
ean
geo
met
ry h
as
a co
rres
pon
din
g st
atem
ent
in s
ph
eric
al g
eom
etry
. If
so, w
rite
th
e co
rres
pon
din
g st
atem
ent.
If
not
, exp
lain
you
r re
ason
ing.
5. A
tri
angl
e ca
n h
ave
at m
ost
one
obtu
se a
ngl
e.
N
o.
A t
rian
gle
can
have a
t m
ost
thre
e o
btu
se a
ng
les.
6. T
he
sum
of
the
angl
es o
f a
tria
ngl
e is
180
°.
N
o.
Th
e s
um
of
the a
ng
les o
f a t
rian
gle
is m
ore
th
an
180°.
7. G
iven
a l
ine
and
a po
int
not
on
th
e li
ne,
th
ere
is e
xact
ly o
ne
lin
e th
at g
oes
thro
ugh
th
e po
int
and
is p
erpe
ndi
cula
r to
th
e li
ne.
Y
es.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
8. A
ll e
quil
ater
al t
rian
gles
are
sim
ilar
.
Y
es.
Th
e s
am
e s
tate
men
t w
ork
s i
n s
ph
eri
cal
geo
metr
y.
9. A
IRPL
AN
ES W
hen
fly
ing
an a
irpl
ane
from
New
Yor
k to
Sea
ttle
, wh
at i
s th
e sh
orte
st
rou
te: f
lyin
g di
rect
ly w
est,
or
flyi
ng
nor
th a
cros
s C
anad
a? E
xpla
in.
T
he s
ho
rtest
rou
te i
s t
o f
ly a
lon
g t
he g
reat
cir
cle
co
nn
ecti
ng
New
Yo
rk
an
d S
eatt
le,
wh
ich
cro
sses C
an
ad
a.
12-7
Prac
tice
Sp
heri
cal
Geo
metr
y
�
⎯
�
CB
an
d �
⎯
�
AE
, �
⎯
�
FA
, �
AD
E
�
⎯
�
ZX
an
d �
⎯
�
TY
, �
⎯
�
ZX
an
d �
TZ
K
N
o
Yes
045_
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OC
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:23
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 12-7
Cha
pte
r 12
47
Gle
ncoe
Geo
met
ry
12-7
Wor
d Pr
oble
m P
ract
ice
Sp
heri
cal
Geo
metr
y
1. P
AIN
TIN
GC
onsi
der
pain
tin
g qu
adri
late
ral
AB
CD
on
th
e be
ach
bal
l w
ith
rad
ius
1 ft
. Wh
at i
s th
e su
rfac
e ar
ea y
ou w
ould
nee
d to
pai
nt?
2. E
AR
THT
he
Equ
ator
an
d th
e P
rim
e M
erid
ian
are
per
pen
dicu
lar
grea
t ci
rcle
s th
at d
ivid
e E
arth
in
to N
orth
, Sou
th a
nd
Eas
t, W
est
hem
isph
eres
. If
Ear
th h
as a
su
rfac
e ar
ea o
f 19
7,00
0,00
0 sq
uar
e m
iles
, wh
at i
s th
e su
rfac
e ar
ea o
f th
e N
orth
-Eas
t se
ctio
n o
f E
arth
?
Sou
rce:
NA
SA
3. O
CEA
NIf
th
e oc
ean
s co
ver
70%
of
Ear
th’s
su
rfac
e, w
hat
is
the
surf
ace
area
of
th
e oc
ean
s?
Sou
rce:
NA
SA
4. G
EOM
ETR
YT
hre
e n
onid
enti
cal
lin
es o
n
the
circ
le d
ivid
e it
in
to e
ith
er
6 se
ctio
ns
or 8
tri
angl
es. W
hat
con
diti
on
is n
eede
d so
th
at t
he
thre
e li
nes
for
m
6 se
ctio
ns?
5. G
EOG
RA
PHY
Lat
itu
de a
nd
lon
gitu
de
lin
es a
re i
mag
inar
y li
nes
on
Ear
th. T
he
lin
es o
f la
titu
de a
re h
oriz
onta
l co
nce
ntr
ic
circ
les
that
hel
p to
def
ine
the
dist
ance
a
plac
e is
fro
m t
he
equ
ator
. Lin
es o
f la
titu
de a
re m
easu
red
in d
egre
es. T
he
equ
ator
is
0°. T
he
nor
th p
ole
is 9
0° n
orth
la
titu
de. T
he
lin
es o
f lo
ngi
tude
are
gre
at
circ
les
that
hel
p to
def
ine
the
dist
ance
a
plac
e is
fro
m t
he
Pri
me
Mer
idan
, wh
ich
is
loc
ated
in
En
glan
d an
d co
nsi
dere
d th
e lo
ngi
tude
of
0°.
a.
Th
e m
ean
rad
ius
of E
arth
is
3963
m
iles
. Atl
anta
, Geo
rgia
, has
co
ordi
nat
es (
33°N
, 84°
W)
and
Cin
cin
nat
i, O
hio
, has
coo
rdin
ates
(3
9°N
, 84°
W).
Est
imat
e th
e di
stan
ce
betw
een
Atl
anta
an
d C
inci
nn
ati
to
the
nea
rest
ten
th.
b
. S
eatt
le, W
ash
ingt
on, h
as c
oord
inat
es
(47°
N, 1
22°W
) an
d P
ortl
and,
Ore
gon
,h
as c
oord
inat
es (
45°N
, 122
°W).
E
stim
ate
the
dist
ance
bet
wee
n
Por
tlan
d an
d S
eatt
le t
o th
e n
eare
st
ten
th.
A C
DB
6π
−
7
sq
uare
feet
2
4,6
25,0
00 s
qu
are
miles
1
37,9
00,0
00 s
qu
are
miles
A
ll t
hre
e l
ines m
ust
inte
rsect
at
two
po
ints
.
4
15.4
miles
138.3
miles
045_
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Answers (Lesson 12-7)
A01-A26_GEOCRMC12_890521.indd A22A01-A26_GEOCRMC12_890521.indd A22 6/2/09 5:32:46 PM6/2/09 5:32:46 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A23 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
4
8
Gle
ncoe
Geo
met
ry
12-7
Pro
jecti
on
sW
hen
mak
ing
map
s of
Ear
th, c
arto
grap
her
s m
ust
sh
ow a
sph
ere
on a
pla
ne.
To
do t
his
th
ey
hav
e to
use
pro
ject
ion
s, a
met
hod
of
con
vert
ing
a sp
her
e in
to a
pla
ne.
Bu
t th
ese
proj
ecti
ons
hav
e th
eir
lim
itat
ion
s.
Th
e m
ap o
n t
he
righ
t is
a M
erca
tor
proj
ecti
on
of E
arth
. On
th
is m
ap G
reen
lan
d ap
pear
s to
be
the
sam
e si
ze a
s A
fric
a. B
ut
Gre
enla
nd
has
a l
and
area
of
2,16
6,08
6 sq
uar
e ki
lom
eter
s an
d A
fric
a h
as a
lan
d ar
ea o
f 30
,365
,700
squ
are
kilo
met
ers.
Th
e m
ap o
n t
he
righ
t is
a L
ambe
rt p
roje
ctio
n.
Wh
en a
pil
ot d
raw
s a
stra
igh
t li
ne
betw
een
tw
o po
ints
on
th
is m
ap t
he
lin
e sh
ows
tru
e be
arin
g, o
r re
lati
ve d
irec
tion
to
the
Nor
th P
ole.
How
ever
, th
e bo
ttom
are
a of
th
is m
ap d
isto
rts
dist
ance
s.
1. W
hen
wou
ld i
t be
use
ful
to u
se a
Mer
cato
r pr
ojec
tion
of
Ear
th?
T
o m
easu
re d
ista
nces c
lose t
o t
he e
qu
ato
r.
2. D
oes
each
squ
are
on t
he
Mer
cato
r pr
ojec
tion
hav
e th
e sa
me
surf
ace
area
? E
xpla
in.
N
o.
Near
the t
op
, th
e w
idth
acro
ss i
s m
uch
sm
aller
than
th
e e
qu
ato
r.
3. D
oes
each
squ
are
on t
he
Lam
bert
pro
ject
ion
hav
e th
e sa
me
surf
ace
area
? E
xpla
in.
N
o.
Lin
es o
f lo
ng
itu
de b
eco
me s
ho
rter
as y
ou
tra
vel
tow
ard
s t
he p
ole
s.
4. T
he
Mer
cato
r pr
ojec
tion
use
s a
cyli
nde
r to
map
Ear
th, w
hil
e th
e L
ambe
rt p
roje
ctio
n u
ses
a co
ne
to m
ap E
arth
. Wh
at o
ther
sh
apes
do
you
th
ink
cou
ld b
e u
sed
to m
ap E
arth
?
T
rian
gle
s,
hexag
on
s,
pri
sm
s.
Enri
chm
ent
Sp
heri
cal
Geo
metr
y
60˚ N
180˚
W18
0˚ E
90˚ W
0˚90
˚ E
40˚ N
20˚ N
0˚ 20˚ S
40˚ S
60˚ S
045_
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Lesson 12-8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
49
Gle
ncoe
Geo
met
ry
Iden
tify
Co
ng
ruen
t o
r Si
mila
r So
lids
Sim
ilar
sol
ids
hav
e ex
actl
y th
e sa
me
shap
e bu
t n
ot n
eces
sari
ly t
he
sam
e si
ze. T
wo
soli
ds a
re s
imil
ar i
f th
ey a
re t
he
sam
e sh
ape
and
the
rati
os o
f th
eir
corr
espo
ndi
ng
lin
ear
mea
sure
s ar
e eq
ual
. All
sph
eres
are
sim
ilar
an
d al
l cu
bes
are
sim
ilar
. Con
gru
ent
soli
ds
hav
e ex
actl
y th
e sa
me
shap
e an
d th
e sa
me
size
. C
ongr
uen
t so
lids
are
sim
ilar
sol
ids
wit
h a
sca
le f
acto
r of
1:1
. Con
gru
ent
soli
ds h
ave
the
foll
owin
g ch
arac
teri
stic
s:
• C
orre
spon
din
g an
gles
are
con
gru
ent
• C
orre
spon
din
g ed
ges
are
con
gru
ent
• C
orre
spon
din
g fa
ces
are
con
gru
ent
• V
olu
mes
are
equ
al
D
eter
min
e w
het
her
th
e p
air
of s
olid
s is
sim
ila
r, c
ongr
uen
t, o
r n
eith
er. I
f th
e so
lid
s ar
e si
mil
ar, s
tate
th
e sc
ale
fact
or.
rati
o of
wid
th:
3 −
6 =
1 −
2
rati
o of
len
gth
: 4 −
8 =
1 −
2
rati
o of
hyp
oten
use
: 5 −
10
= 1 −
2
rati
o of
hei
ght:
4 −
8 =
1 −
2
Th
e ra
tios
of
the
corr
espo
ndi
ng
side
s ar
e eq
ual
, so
the
tria
ngu
lar
pris
ms
are
sim
ilar
. Th
e sc
ale
fact
or i
s 1:
2. S
ince
th
e sc
ale
fact
or i
s n
ot 1
:1, t
he
soli
ds a
re n
ot c
ongr
uen
t.
Exer
cise
sD
eter
min
e w
het
her
th
e p
air
of s
olid
s is
sim
ila
r, c
ongr
uen
t, o
r n
eith
er. I
f th
e so
lid
s ar
e si
mil
ar, s
tate
th
e sc
ale
fact
or.
1.
2 cm
1 cm
10 c
m
5 cm
2.
4.2
in.
12.3
in.
4.2
in.
12.3
in.
3.
4 in
.
8 in
.
4.
2 m
2 m
4 m
1 m
1 m
3 m
Stud
y G
uide
and
Inte
rven
tion
Co
ng
ruen
t an
d S
imilar
So
lid
s
12-8
5 in
.
4 in
.
4 in
.3
in.
10 in
.
8 in
.
8 in
.
6 in
.Ex
amp
le
S
imilar,
1:5
C
on
gru
en
t
S
imilar,
1:2
N
eit
her
045_
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M
Answers (Lesson 12-7 and Lesson 12-8)
A01-A26_GEOCRMC12_890521.indd A23A01-A26_GEOCRMC12_890521.indd A23 6/1/09 12:12:50 PM6/1/09 12:12:50 PM
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A24 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
5
0
Gle
ncoe
Geo
met
ry
Pro
per
ties
of
Co
ng
ruen
t o
r Si
mila
r So
lids
Wh
en p
airs
of
soli
ds a
re c
ongr
uen
t or
si
mil
ar, c
erta
in p
rope
rtie
s ar
e kn
own
.
If t
wo
sim
ilar
sol
ids
hav
e a
scal
e fa
ctor
of
a:b
then
,•
the
rati
o of
th
eir
surf
ace
area
s is
a2 :b
2 .•
the
rati
o of
th
eir
volu
mes
is
a3 :b3 .
T
wo
sph
eres
hav
e ra
dii
of
2 fe
et a
nd
6 f
eet.
W
hat
is
the
rati
o of
th
e vo
lum
e of
th
e sm
all
sph
ere
to t
he
volu
me
of t
he
larg
e sp
her
e?
Fir
st, f
ind
the
scal
e fa
ctor
.
radi
us
of t
he
smal
l sp
her
e
−
−
rad
ius
of t
he
larg
e sp
her
e = 2 −
6 o
r 1 −
3
Th
e sc
ale
fact
or i
s 1 −
3 .
a3 −
b3
= (1
)3 −
(3
)3 or
1 −
27
So,
th
e ra
tio
of t
he
volu
mes
is
1:27
.
Exer
cise
s 1
. Tw
o cu
bes
hav
e si
de l
engt
hs
of 3
in
ches
an
d 8
inch
es. W
hat
is
the
rati
o of
th
e su
rfac
e ar
ea o
f th
e sm
all
cube
to
the
surf
ace
area
of
the
larg
e cu
be?
9
:64
2. T
wo
sim
ilar
con
es h
ave
hei
ghts
of
3 fe
et a
nd
12 f
eet.
Wh
at i
s th
e ra
tio
of t
he
volu
me
of
the
smal
l co
ne
to t
he
volu
me
of t
he
larg
e co
ne?
1
:64
3. T
wo
sim
ilar
tri
angu
lar
pris
ms
hav
e vo
lum
es o
f 27
squ
are
met
ers
and
64 s
quar
e m
eter
s.
Wh
at i
s th
e ra
tio
of t
he
surf
ace
area
of
the
smal
l pr
ism
to
the
surf
ace
area
of
the
larg
e pr
ism
?
9
:16
4. C
OM
PUTE
RS
A s
mal
l re
ctan
gula
r la
ptop
has
a w
idth
of
10 i
nch
es a
nd
an a
rea
of
80 s
quar
e in
ches
. A l
arge
r an
d si
mil
ar l
apto
p h
as a
wid
th o
f 15
in
ches
. Wh
at i
s th
e le
ngt
h o
f th
e la
rger
lap
top?
1
2 i
n.
5. C
ON
STR
UC
TIO
N A
bu
ildi
ng
com
pan
y u
ses
two
sim
ilar
siz
es o
f pi
pes.
Th
e sm
alle
r si
ze
has
a r
adiu
s of
1 i
nch
an
d le
ngt
h o
f 8
inch
es. T
he
larg
er s
ize
has
a r
adiu
s of
2.5
in
ches
W
hat
is
the
volu
me
of t
he
larg
er p
ipes
?
1
25
π c
ub
ic i
n.
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nu
ed)
Co
ng
ruen
t an
d S
imilar
So
lid
s
Exam
ple
12-8
2 ft
6 ft
045_
055_
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Lesson 12-8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
51
Gle
ncoe
Geo
met
ry
Det
erm
ine
wh
eth
er e
ach
pai
r of
sol
ids
is s
imil
ar,
con
gru
ent,
or
nei
ther
. If
the
soli
ds
are
sim
ilar
, sta
te t
he
scal
e fa
ctor
.
1.
2 cm
4 cm3
cm
6 cm
12 c
m
9 cm
2.
6 cm
9 cm
8 cm
12 c
m
3.
5 m
10 m
4.
3 ft
1 ft
1 ft
9 ft
3 ft
3 ft
5. T
wo
sim
ilar
pyr
amid
s h
ave
hei
ghts
of
4 in
ches
an
d 7
inch
es W
hat
is
the
rati
o of
th
e vo
lum
e of
th
e sm
all
pyra
mid
to
the
volu
me
of t
he
larg
e py
ram
id?
6
4:3
43
6. T
wo
sim
ilar
cyl
inde
rs h
ave
surf
ace
area
s of
40π
squ
are
feet
an
d 90
π s
quar
e fe
et. W
hat
is
th
e ra
tio
of t
he
hei
ght
of t
he
larg
e cy
lin
der
to t
he
hei
ght
of t
he
smal
l cy
lin
der?
3
:2
7. C
OO
KIN
G T
wo
stoc
kpot
s ar
e si
mil
ar c
ylin
ders
. Th
e sm
alle
r st
ockp
ot h
as a
hei
ght
of
10 i
nch
es a
nd
a ra
diu
s of
2.5
in
ches
. Th
e la
rger
sto
ckpo
t h
as a
hei
ght
of 1
6 in
ches
. Wh
at
is t
he
volu
me
of t
he
larg
er s
tock
pot?
Rou
nd
to t
he
nea
rest
ten
th.
Skill
s Pr
acti
ceC
on
gru
en
t an
d S
imilar
So
lid
s
12-8
s
imilar;
1:3
s
imilar;
3:4
c
on
gru
en
t s
imilar;
1:3
804.2
in
3
045_
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Answers (Lesson 12-8)
A01-A26_GEOCRMC12_890521.indd A24A01-A26_GEOCRMC12_890521.indd A24 6/3/09 11:24:43 PM6/3/09 11:24:43 PM
An
swer
s
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 12 A25 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
5
2
Gle
ncoe
Geo
met
ry
Det
erm
ine
wh
eth
er t
he
pai
r of
sol
ids
is s
imil
ar,
con
gru
ent,
or
nei
ther
. If
the
soli
ds
are
sim
ilar
, sta
te t
he
scal
e fa
ctor
.
1.
6 cm
8 cm
18 c
m
24 c
m
2.
5 cm
12 c
m
10 c
m
24 c
m
3.
1 m
1 m
5 m
5 m
3 m
3 m
4 m
4 m
4.
5 cm
5 cm
2 cm
10 c
m
10 c
m
1.5
cm
5. T
wo
cube
s h
ave
surf
ace
area
s of
72
squ
are
feet
an
d 98
squ
are
feet
. Wh
at i
s th
e ra
tio
of
the
volu
me
of t
he
smal
l cu
be t
o th
e vo
lum
e of
th
e la
rge
cube
?
2
16:3
43
6. T
wo
sim
ilar
ice
cre
am c
ones
are
mad
e of
a h
alf
sph
ere
on t
op a
nd
a co
ne
on b
otto
m.
Th
ey h
ave
radi
i of
1 i
nch
an
d 1.
75 i
nch
es r
espe
ctiv
ely.
Wh
at i
s th
e ra
tio
of t
he
volu
me
of t
he
smal
l ic
e cr
eam
con
e to
th
e vo
lum
e of
th
e la
rge
ice
crea
m c
one?
Rou
nd
to t
he
nea
rest
ten
th.
1
:5.4
7. A
RC
ITH
ECTU
RE
Arc
hit
ects
mak
e sc
ale
mod
els
of b
uil
din
gs t
o pr
esen
t th
eir
idea
s to
cl
ien
ts. I
f an
arc
hit
ect
wan
ts t
o m
ake
a 1:
50 s
cale
mod
el o
f a
4000
squ
are
foot
hou
se,
how
man
y sq
uar
e fe
et w
ill
the
mod
el h
ave?
1
.6 s
qu
are
feet
12-8
Prac
tice
Co
ng
ruen
t an
d S
imilar
So
lid
s
s
imilar,
1:3
n
eit
her
c
on
gru
en
t n
eit
her
045_
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M
Lesson 12-8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
53
Gle
ncoe
Geo
met
ry
12-8
Wor
d Pr
oble
m P
ract
ice
Co
ng
ruen
t an
d S
imilar
So
lid
s
1. C
OO
KIN
GA
cyl
indr
ical
pot
is
4.5
inch
esta
ll a
nd
has
a r
adiu
s of
4 i
nch
es. H
ow
tall
wou
ld a
sim
ilar
pot
be
if i
ts r
adiu
s is
6
inch
es?
2. M
AN
UFA
CTU
RIN
GB
oxes
, In
c. w
ants
to
mak
e th
e tw
o bo
xes
belo
w. H
ow l
ong
does
th
e se
con
d bo
x n
eed
to b
e so
th
at
they
are
sim
ilar
?
24 c
m
15 c
m25
cm
25 c
m15
cm
3. F
AR
MIN
GA
far
mer
has
tw
o si
mil
ar
cyli
ndr
ical
gra
in s
ilos
. Th
e sm
alle
r si
lo
is 2
5 fe
et t
all
and
the
larg
er s
ilo
is
40 f
eet
tall
. If
the
smal
ler
silo
can
hol
d 15
00 c
ubi
c fe
et o
f gr
ain
, how
mu
ch c
an
the
larg
er s
ilo
hol
d?
4. P
LAN
ETS
Ear
th h
as a
su
rfac
e ar
ea o
f ab
out
196,
937,
500
squ
are
mil
es. M
ars
has
a s
urf
ace
area
of
abou
t 89
,500
,000
sq
uar
e m
iles
. Wh
at i
s th
e ra
tio
of t
he
radi
us
of E
arth
to
the
radi
us
of M
ars?
R
oun
d to
th
e n
eare
st t
enth
.
Sou
rce:
NA
SA
5. B
ASE
BA
LLM
ajor
Lea
gue
Bas
ebal
l or
M
LB
, ru
les
stat
e th
at b
aseb
alls
mu
st
hav
e a
circ
um
fere
nce
of
9 in
ches
. Th
e N
atio
nal
Sof
tbal
l A
ssoc
iati
on, o
r N
SA
, ru
les
stat
e th
at s
oftb
alls
mu
st h
ave
a ci
rcu
mfe
ren
ce n
ot e
xcee
din
g 12
in
ches
.
Sou
rce:
ML
B, N
SA
a.
Fin
d th
e ra
tio
of t
he
circ
um
fere
nce
of
ML
B b
aseb
alls
to
the
circ
um
fere
nce
of
NS
A s
oftb
alls
.
b
. F
ind
the
rati
o of
th
e vo
lum
e of
ML
B
base
ball
s to
th
e vo
lum
e of
NS
A
soft
ball
s. R
oun
d to
th
e n
eare
st t
enth
.
6
.75 i
n.
4
0 c
m
6
144 c
ub
ic f
eet
of
gra
in
1
.5:1
3
:4
2
7:6
4
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Answers (Lesson 12-8)
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Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 12 A26 Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Cha
pte
r 12
5
4
Gle
ncoe
Geo
met
ry
Enri
chm
ent
Do
ub
lin
g S
izes
Con
side
r w
hat
hap
pen
s to
su
rfac
e ar
ea w
hen
th
e si
des
of a
fig
ure
are
dou
bled
.
Th
e si
des
of t
he
larg
e cu
be a
re t
wic
e th
e si
ze o
f th
e si
des
of t
he
sm
all
cube
.
1. H
ow l
ong
are
the
edge
s of
th
e la
rge
cube
?
2. W
hat
is
the
surf
ace
area
of
the
smal
l cu
be?
3. W
hat
is
the
surf
ace
area
of
the
larg
e cu
be?
4. T
he
surf
ace
area
of
the
larg
e cu
be i
s h
ow m
any
tim
es g
reat
er
than
th
at o
f th
e sm
all
cube
?
Th
e ra
diu
s of
th
e la
rge
sph
ere
at t
he
righ
t is
tw
ice
the
radi
us
of
the
smal
l sp
her
e.
5. W
hat
is
the
surf
ace
area
of
the
smal
l sp
her
e?
6. W
hat
is
the
surf
ace
area
of
the
larg
e sp
her
e?
7. T
he
surf
ace
area
of
the
larg
e sp
her
e is
how
man
y ti
mes
gr
eate
r th
an t
he
surf
ace
area
of
the
smal
l sp
her
e?
8. I
t ap
pear
s th
at i
f th
e di
men
sion
s of
a s
olid
are
dou
bled
, th
e su
rfac
e ar
ea i
s m
ult
ipli
ed b
y .
Now
con
side
r h
ow d
oubl
ing
the
dim
ensi
ons
affe
cts
the
volu
me
of a
cu
be.
Th
e si
des
of t
he
larg
e cu
be a
re t
wic
e th
e si
ze o
f th
e si
des
of t
he
smal
l cu
be.
9. H
ow l
ong
are
the
edge
s of
th
e la
rge
cube
?
10. W
hat
is
the
volu
me
of t
he
smal
l cu
be?
11. W
hat
is
the
volu
me
of t
he
larg
e cu
be?
12. T
he
volu
me
of t
he
larg
e cu
be i
s h
ow m
any
tim
es g
reat
er t
han
th
at o
f th
e sm
all
cube
?
Th
e la
rge
sph
ere
at t
he
righ
t h
as t
wic
e th
e ra
diu
s of
th
e sm
all
sph
ere.
13. W
hat
is
the
volu
me
of t
he
smal
l sp
her
e?
14. W
hat
is
the
volu
me
of t
he
larg
e sp
her
e?
15. T
he
volu
me
of t
he
larg
e sp
her
e is
how
man
y ti
mes
gre
ater
th
an t
he
volu
me
of t
he
smal
l sp
her
e?
16. I
t ap
pear
s th
at i
f th
e di
men
sion
s of
a s
olid
are
dou
bled
, th
e vo
lum
e is
mu
ltip
lied
by
.
5 in
.
3 m
5 in
.
3 m
12-8
10 i
n.
150 i
n2
600 i
n2
4 t
imes
36
π m
2
144
π m
2
4 t
imes
4 10 i
n.
125 i
n3
1000 i
n3
8 t
imes
36
π m
3
288
π m
3
8 t
imes
8
045_
055_
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M
Answers (Lesson 12-8)
A01-A26_GEOCRMC12_890521.indd A26A01-A26_GEOCRMC12_890521.indd A26 6/14/08 9:22:07 PM6/14/08 9:22:07 PM
Copyright
© G
lencoe/M
cG
raw
-Hill
, a d
ivis
ion o
f T
he M
cG
raw
-Hill
Com
panie
s,
Inc.
An
swer
s
Chapter 12 A27 Glencoe Geometry
PDF Pass
Chapter 12 Assessment Answer KeyQuiz 1 (Lessons 12-1 and 12-2) Quiz 3 (Lessons 12-4 and 12-5) Mid-Chapter TestPage 57 Page 58 Page 59
Quiz 4 (Lessons 12-7 and 12-8) Page 58
1.
2.
3.
4.
5.
116.1 in2
188.5 ft2
267.0 ft2
1152 ft3
2827.4 in3
1.
2.
3.
4.
5.
156 cm3
1005.3 cm3
261.8 cm3
201.1 m2
C
1.
2.
3.
4.
5.
� ⎯⎯ � WX and � ⎯ � VU , −−
ZX,
and � RSZ
yes
similar, 2:1
27:125
27:64
1.
2.
3.
4.
5.
C
G
D
F
D
6.
7.
8.
9.
10.
92.8 units2
2884.0 in2
150.8 in2
1.
2.
3.
4.
5.
8.6 m
120 units2
360 cm2
B
Quiz 2 (Lessons 12-3 and 12-4)
Page 57
A27-A36_GEOCRMC12_890521.indd 27A27-A36_GEOCRMC12_890521.indd 27 6/1/09 12:26:45 PM6/1/09 12:26:45 PM
Copyrig
ht ©
Gle
ncoe/M
cG
raw
-Hill, a
div
isio
n o
f Th
e M
cG
raw
-Hill C
om
panie
s, In
c.
Chapter 12 A28 Glencoe Geometry
PDF Pass
Chapter 12 Assessment Answer KeyVocabulary Test Form 1 Page 60 Page 61 Page 62
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
slant height
similar solids
right cylinder
volume
axis
congruent solids
false; pyramid
true
true
false; oblique cone
the sum of the areas of the lateral
faces
a prism whose lateral edges are
also altitudes
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
C
F
B
G
D
H
A
F
B
H
B
12.
13.
14.
15.
16.
17.
18.
19.
20.
G
D
H
D
F
B
G
B
J
B: 1020 ft2
A27-A36_GEOCRMC12_890521.indd 28A27-A36_GEOCRMC12_890521.indd 28 6/14/08 3:04:56 PM6/14/08 3:04:56 PM
Copyright
© G
lencoe/M
cG
raw
-Hill
, a d
ivis
ion o
f T
he M
cG
raw
-Hill
Com
panie
s,
Inc.
An
swer
s
Chapter 12 A29 Glencoe Geometry
Chapter 12 Assessment Answer KeyForm 2A Form 2B Page 63 Page 64 Page 65 Page 66
12.
13.
14.
15.
16.
17.
18.
19.
20.
H
C
J
B
G
B
H
C
G
B: 53 ft2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
D
H
D
G
C
F
C
F
B
G
C
12.
13.
14.
15.
16.
17.
18.
19.
20.
H
A
H
B
F
D
F
B
H
B: 391.6 ft2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
J
A
G
D
H
A
H
A
G
B
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Chapter 12 A30 Glencoe Geometry
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Chapter 12 Assessment Answer KeyForm 2C Page 67 Page 68
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
2 ft
7690.6 ft3
467.7 ft3
103.2 ft3
1140.4 cm2
208.8 in3
great circle
neither
612.5 m2
113.1 in2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
�PQRS, �PQT,
�QTR, �RTS, �PTS
120 cm2
6 in2
524.1 yd2
6 in.
270 in2
270 + 150 √ � 3 in2
62.8 ft2
113.1 ft2
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An
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Chapter 12 A31 Glencoe Geometry
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Chapter 12 Assessment Answer KeyForm 2D Page 69 Page 70
12.
13.
14.
15.
16.
17.
18.
19.
576 in3
923.6 cm3
2057 in3
100.5 cm3
1592.8 in2
106.4 cm3
similar
1562.5 ft3
Lines in spherical
geometry are great
circles. Great circles
always intersect,
therefore they cannot be
parallel.
20.
B: 201.1 in2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
−−
GJ , −−
HJ , −
IJ , −−
GH , −−
HI , −−
GI
750 in2
258 ft2
845.2 m2
7 ft
540 ft2
931.0 ft2
204.2 cm2
282.7 cm2
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Chapter 12 A32 Glencoe Geometry
Chapter 12 Assessment Answer KeyForm 3 Page 71 Page 72
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
It is a rectangle.
128 + 4 √ �� 21
units2
35 + 7 √ �� 13 ft2
528 units2
17,837.8 ft2
32,435.2 ft2
468 in2
842.1 in2
233.8 in2
387.7 in2
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
184 cm3
8 m
No, it will overflow.
148.4 in3 > 96 in3
121.5 in3
3πr2
Yes; 33.5 cm3
< 37.7 cm3
similar
1: 3 √ � 4
700 ft2
No. Any three points
on a sphere determine
a triangle.
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Chapter 12 A33 Glencoe Geometry
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Chapter 12 Assessment Answer KeyExtended-Response Test, Page 73
Scoring Rubric
Score General Description Specifi c Criteria
4 Superior
A correct solution that
is supported by well-developed,
accurate explanations
• Shows thorough understanding of the concepts of pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.
• Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• Figures and drawings are accurate and appropriate.
• Goes beyond requirements of some or all problems.
3 Satisfactory
A generally correct solution, but may
contain minor fl aws in reasoning or
computation
• Shows an understanding of the concepts of pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.
• Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Figures and drawings are mostly accurate and appropriate.
• Satisfi es all requirements of problems.
2 Nearly Satisfactory
A partially correct interpretation and/or
solution to the problem
• Shows an understanding of most of the concepts of
pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Figures and drawings are mostly accurate.
• Satisfi es the requirements of most of the problems.
1 Nearly Unsatisfactory
A correct solution with no supporting
evidence or explanation
• Final computation is correct.
• No written explanations or work shown to substantiate the
fi nal computation.
• Figures and drawings may be accurate but lack detail or
explanation.
• Satisfi es minimal requirements of some of the problems.
0 Unsatisfactory
An incorrect solution indicating no
mathematical understanding of the
concept or task, or no solution is given
• Shows little or no understanding of most of the concepts of
pyramids, prisms, cylinders, cones, spheres, surface area, lateral area, volume, and properties of solid fi gures.
• Does not use appropriate strategies to solve problems.
• Computations are incorrect.
• Written explanations are unsatisfactory.
• Figures and drawings are inaccurate or inappropriate.
• Does not satisfy requirements of problems.
• No answer given.
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Chapter 12 A34 Glencoe Geometry
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Chapter 12 Assessment Answer KeyExtended-Response Test, Page 73
Sample Answers
1. The lateral area is the area of the lateral faces. The surface area includes the area of the lateral faces plus the areas of the two bases.
2.
Oblique Right
3. Sample answer: Sam is painting the walls of a room. The room is 12 feet long, 10 feet wide, and 8 feet high. A gallon of paint covers 400 square feet and costs $16 per gallon. Find the cost of the paint needed to paint the room.
4. The first cylinder could have a radius of 3, a height of 4, a volume of 36π cubic units, and a surface area of 42π square units. The second cylinder could have a radius of 1, a height of 30, a volume of 30π cubic units, and a surface area of 62π square units.
5.
4 in.4 in.4 in.
12 in.
4 in.
4 in.
The volume of the pyramid is 16 • 12 − 3 or 64 cubic units and
the volume of the prism is 4 • 4 • 4 or 64 cubic units.
6. The volume of the cylinder is πr2 • 2r.
The volume of the hemisphere is 2πr3 −
3 .
The volume of the cone is πr2 • r − 3
Therefore, the total volume is
2πr3 + 2πr3
− 3 + πr3
− 3 or 3πr3 cubic units.
In addition to the scoring rubric found on page A33, the following sample answers may be used as guidance in evaluating open-ended assessment items.
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Chapter 12 A35 Glencoe Geometry
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Chapter 12 Assessment Answer KeyStandardized Test Practice Page 74 Page 75
1.
2. F G H J
3. A B C D
4. F G H J
5.
6. F G H J
7. A B C D
8. F G H J
A B C D
A B C D
9.
10.
11.
12. F G H J
A B C D
F G H J
A B C D
13.
14. 9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
1 4
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
5
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Chapter 12 A36 Glencoe Geometry
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Chapter 12 Assessment Answer KeyStandardized Test Practice Page 76
15.
16.
17.
18.
19.
20.
21a.
b.
c.
d.
2.6 m
25
38.5 cm2
pentagonal
pyramid
3141.6 in2
2206.2 cm2
13,684.8 m2
207.3 m
3421.2 m2
10,263.6 m2
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