11-1 Space Figures and Cross Sections · Using Formulas to Find Surface Area of a Prism ... Complete the flow chart below. 15. The lateral area of the prism is m2. Finding Surface

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Vocabulary

Chapter 11 282

11-1 Space Figures and Cross Sections

Review

Complete each statement with the correct word from the list.

edge edges vertex vertices

1. A(n) 9 is a segment that is formed by the intersections of two faces.

2. A(n) 9 is a point where two or more edges intersect.

3. A cube has eight 9.

4. A cube has twelve 9.

Vocabulary Builder

polyhedron (noun) pahl ih HEE drun (plural: polyhedra)

Related Words: face, edge, vertex

Definition: A polyhedron is a space figure, or three-dimensional figure, whose surfaces are polygons.

Origin: The word polyhedron combines the Greek prefix poli-, meaning “many,” and hedron, meaning “base.”

Examples: prism, pyramid

Non-Examples: circle, cylinder, sphere

Use Your Vocabulary

5. Cross out the figure below that is NOT a polyhedron.

polyhedra

Pyramid Prism

edge

vertex

vertices

edges

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d.Problem 1

Problem 2

Key Concept Euler’s Formula

R

S

U V

T W

283 Lesson 11-1

Identifying Vertices, Edges, and Faces

Got It? How many vertices, edges, and faces are in the polyhedron at the right? List them.

6. Identify each description as a vertex, an edge, or a face.

a point where three or polygon a segment where two

more edges intersect or more faces intersect

7. List the vertices.

8. List the edges. Remember to list the dashed hidden edges.

9. List the faces. Remember to list the hidden faces.

10. The polyhedron has vertices, edges, and faces.

Using Euler’s Formula

Got It? Use Euler’s Formula to find the number of faces for a polyhedron with 30 edges and 20 vertices.

11. Use the justifications at the right to find the number of faces.

F 1 V 5 E 1 2 Use Euler’s Formula.

F 1 5 1 2 Substitute with given information.

F 1 5 Simplify.

F 5 2 Subtraction Property of Equality

F 5 Simplify.

12. A polygon with 30 edges and 20 vertices has faces.

Th e sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number of its edges (E).

F 1 V 5 E 1 2

vertex face edge

6

20

20

32

12

30

32

20

9 5

R, S, T, U, V, W

RS, VW, RU, RV, ST, SW, TU, TW, UV

quadrilateral RSTU, quadrilateral RSWV , quadrilateral TUVW, kSTW , kRUV

12

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Problem 3

Problem 4

Problem 5

Chapter 11 284

Verifying Euler’s Formula in Two Dimensions

Got It? Use the solid at the right. How can you verify Euler’s Formula F 1 V 5 E 1 2 for the solid?

13. Count the number of vertices.

on the bottom 1 on the top 5 vertices

14. Count the number of faces.

bases 1 lateral faces 5 faces

15. Count the number of edges.

solid edges 1 dashed hidden edges 5 edges

16. Now verify Euler’s Formula for the values you found.

F 1 V 5 E 1 2 Write Euler’s Formula.

1 5 1 2 Substitute.

5 Simplify.

Describing a Cross Section

Got It? For the solid at the right, what is the cross section formed by a horizontal plane?

Underline the correct word to complete each sentence.

17. A horizontal plane is parallel to the bottom / side of the solid.

18. A view from the side / top of the solid helps you see the shape of the cross section.

19. The cross section is a circle / trapezoid .

Drawing a Cross Section

Got It? Draw the cross section formed by a horizontal plane intersecting the left and right faces of the cube. What shape is the cross section?

20. A horizontal plane is parallel to which faces of the cube? Circle your answer.

front and back left and right top and bottom

21. Circle the diagram that shows the intersection of the horizontal plane and the left and right faces of the cube.

4

2

9

6 8 12

1414

3 12

4 6

4 8

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Lesson Check

Math Success

KnowTh e octagons and squares are 9.

An edge is a segment formed by the intersection of two 9.

To fi nd the number of 9 without using Euler’s Formula

Count the number of edges formed by the squares and the top octagon.

Count the number of edges formed by the squares and the bottom octagon.

Count the number of edges formed by the squares.

Need Plan

285 Lesson 11-1

Check off the vocabulary words that you understand.

polyhedron face edge vertex cross section

Rate how well you can recognize polyhedra and their parts.

Vocabulary Suppose you build a polyhedron from two octagons and eight squares. Without using Euler’s Formula, how many edges does the solid have? Explain.

24. Complete the problem-solving model below.

25. The intersections of the squares and the top octagon form edges.

26. The intersections of the squares and the bottom octagon form edges.

27. The intersection of the eight squares form edges.

28. The solid has 1 1 5 edges.

• Do you UNDERSTAND?

22. Use the cube to draw and shade the cross section.

23. The cross section is a 9

square

facesedges

faces

8

8

8

24888

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Vocabulary

Chapter 11 286

11-2 Surface Areas of Prisms and Cylinders

Review

Write T for true or F for false.

1. A lateral face is a polygon surface of a solid.

2. Lateral faces are surfaces of a polyhedron.

3. A lateral face may be a circle.

4. A base is a lateral face.

Vocabulary Builder

oblique (adjective) oh BLEEK

Definition: An oblique object is slanting, not straight.

Main Idea: Oblique means indirect and not straight to the point.

Other Word Forms: obliquely (adverb)

Math Usage: An oblique polyhedron has no vertical edge so an oblique prism is not a right prism.

Use Your Vocabulary

5. Circle the oblique prism. 6. Circle the oblique cylinder.

7. Complete with oblique or obliquely.

A right prism is not an 9 prism.

Your classmate answered the question 9 .

T

T

F

F

oblique

obliquely

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d.Problem 1

Theorem 11-1 Lateral and Surface Areas of a Prism

5 cm

6 cm

5 cm

12 cm

5 cm

hh

B is the area of a base.

p is the perimeter of a base.

5

12

6

55

5 D

E

A B C

287 Lesson 11-2

Using a Net to Find Surface Area of a Prism

Got It? What is the surface area of the triangular prism? Use a net.

8. Label the missing dimensions in the net below.

9. The altitude a of each triangle forms a right triangle with legs of lengths a cm and cm.

10. Use the Pythagorean Theorem to find a.

21 a2 5 52

a2 5 25 2

a2 5

a 5

11. Find the surface area of the prism.

S.A. 5 L.A. 1 area of base

5 areas of two lateral rectangles 1 areas of two lateral triangles 1 area of base

5 (Area A 1 Area C) 1 (Area D 1 Area E) 1 Area B

5 5 ? 1 5 ? 112( ? ) 1 1

2( ? ) 1 ?

5 1 1 1 1

5

12. The surface area of the triangular prism is cm2.

Th e lateral area of a right prism is the product of the perimeter of the base and the height of the prism.

L.A. 5 ph

Th e surface area of a right prism is the sum of the lateral area and the areas of the two bases.

S.A. 5 L.A. 1 2B

13. Write the formula for S.A. using p and h.

S.A. 5 1 2B

3

3

9

16

12

60

ph

216

216

12 12 7260

12 6 4 6 4 6 12

4

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p perimeter of base

6 6 36L.A. 36 12

432h height of prism

12

L.A. ph

Problem 2

WriteThink

I can use the formula for the surface

area of a cylinder.

Then I can substitute the formulas for

lateral area and area of a circle.

Now I simplify.

Next I substitute 10 for the radius and

9 for the height.

S.A. L.A. 2B

2 rh r2 2

180 200

380

10 9 102 2 2

Problem 3

6 m

12 m

h

r

B is the area of a base.

Chapter 11 288

Using Formulas to Find Surface Area of a Prism

Got It? What is the lateral area of the prism at the right?

14. Complete the flow chart below.

15. The lateral area of the prism is m2.

Finding Surface Area of a Cylinder

Got It? A cylinder has a height of 9 cm and a radius of 10 cm. What is the surface area of the cylinder in terms of π?

17. Use the information in the problem to complete the reasoning model below.

18. The surface area of the cylinder is pcm2.

Theorem 11–2 Lateral and Surface Areas of a Cylinder

16. Use the diagram at the right to complete the formulas below.

The lateral area of a right cylinder is The surface area of a right cylinderthe product of the circumference of is the sum of the lateral area andthe base and the height of the cylinder. the areas of the two bases.

L.A. 5 2pr ? h, or L.A. 5 p ? ? h S.A. 5 L.A. 1 2B

5 1 2pr 2

432

380

d

πdh

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Lesson Check

Problem 4

Math Success

CongruentFaces

andFaces

||ParallelFaces

ABFEDCGH

ABCDEFGH

BFGCAEHD

A

B

D

H

GF

CE

289 Lesson 11-2


right prism oblique prism right cylinder oblique cylinder

Rate how well you can fi nd the surface area of a prism and a cylinder.

Vocabulary Name the lateral faces and the bases of the prism at the right.

22. Write the name of each of the faces of the prism in the correct region of the Venn diagram.

23. Name the bases of the prism. 24. Name the lateral faces of the prism.


Finding Lateral Area of a Cylinder

Got It? A stencil roller has a height of 1.5 in. and a diameter of 2.5 in. What area does the roller cover in one turn? Round your answer to the nearest tenth.

19. Underline the correct words to complete the sentence. The distance that is covered

in one turn is the circumference / diameter of the circular base of the cylinder / prism .

20. Find the area the roller covers in one turn.

21. The roller covers about in.2 in one turn.

ABCD and EFGH ABFE, DCGH, BFGC, AEHD

L.A. 5 πdh 5 π(2.5)(1.5) N 11.780972

6

11.8

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Vocabulary

5 m 5 m 5 m

2 in. 3 in.

Chapter 11 290

11-3 Surface Areas of Pyramids and Cones

Review

Label each diagram cone or pyramid.

1. 2. 3. 4.

Vocabulary Builder

slant height (noun) slant hyt

Related Words: regular pyramid, lateral face

Definition: The slant height / of a regular pyramid is the length of the altitude of a lateral face of the pyramid. The slant height / of a cone is the distance from the vertex of a cone to a point on the circumference of the base.

Math Usage: The slant height of a regular pyramid divides the lateral face into two congruent right triangles.

Use Your Vocabulary

5. Circle the figure that shows a three-dimensional figure with slant height 5 m.

6. Is the slant height the same as the height of a pyramid or cone? Yes / No

7. The slant height of the cone at the right is in.

slant height, <

cone pyramid

3

pyramid cone

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Problem 1

Theorem 11-3 Lateral and Surface Areas of a Pyramid

B

291 Lesson 11-3

Finding the Surface Area of a Pyramid

Got It? A square pyramid has base edges of 5 m and a slant height of 3 m. What is the surface area of the pyramid?


13. Find p. 14. Find B.

p 5 4(s) B 5 s2

5 4( ) 5 2

5 5

15. L.A. 5 12p/ 16. S.A. 5 L.A. 1 B

512 ( )( ) 5 1

5 5

17. The surface area of the pyramid is m2.

Th e lateral area (L.A.) of a regular pyramid is half the product of the perimeter p of the base and the slant height / of the pyramid.

Th e surface area (S.A.) of a regular pyramid is the sum of the lateral area and the area B of the base.

8. In a square pyramid with base side length s, p 5 .

9. If the base of a regular pyramid has a perimeter of 6q and its side length

is 6, the pyramid has sides.

Draw a line from each description in Column A to the corresponding formula in Column B.

Column A Column B

10. lateral area (L.A.) of a pyramid 12p/

11. surface area (S.A.) of a pyramid 12p/ 1 B

KnowTh e base is a square with

side length m.

Slant height / is m.

Lateral area

Surface area

Find the perimeter of the base.

Find the area of the base.

Use p and / to fi nd

Use L.A. and B to fi nd

Need Plan

5

3

5

20

20 3

30 55

55

30 25

25

5

p

B

L.A.

S.A.

4s

q

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Problem 2

Problem 3

42 ft

ft18 3

42 ft

36 ft

18 3 ft

Base

Altitudeh

r

VertexSlantheight

Chapter 11 292

Finding the Lateral Area of a Pyramid

Got It? What is the lateral area of the hexagonal pyramid at the right? Round to the nearest square foot.

18. Circle the correct equation for the perimeter of the hexagonal base.

42 ? 36 5 1512 6 ? 18 5 108 12(36)(18!3) < 561 6 ? 36 5 216

19. The slant height / of the pyramid is the hypotenuse of a right triangle. Label the legs of the right triangle at the right.

20. Use the Pythagorean Theorem to find the slant height / of the pyramid.

/ 5 Å 2 1 2

5 Å 1

5 Å

21. Use the formula L.A. 5 12 p/ to find the lateral area of the pyramid.

L.A. 5 12 ( )( ) <

22. The lateral area of the hexagonal pyramid is about ft2.

Finding the Surface Area of a Cone

Got It? The radius of the base of a cone is 16 m. Its slant height is 28 m. What is the surface area in terms of π?

24. Use the justifications at the right to find the surface area.

S.A. 5 L.A. 1 B Use the formula for surface area.

5 1 Substitute the formulas for L.A. and B.

5 p( )( ) 1 p( )2 Substitute for r and for /.

5 p( ) 1 p( ) Simplify.

5 p( ) Add.

25. The surface area of the cone in terms of p is m2.

Th e lateral area of a right cone is half the product of the circumference of the base and the slant height of the cone.

L.A. 5 12 ? 2pr ? /, or L.A. 5 pr/

Th e surface area of a cone is the sum of the lateral area and the area of the base.

23. S.A. 5 L.A. 1

Theorem 11-4 Lateral and Surface Areas of a Cone

(18!3)

972

2736

1764

"2736216

B

πr<

16

448

704

704π

256

28 16

πr2

5649

5649.133031

42

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Lesson Check

Problem 4

Math Success

28 in.

10 in.

A. Base is a polygon.

B. Faces are rectangles.

C. Faces are triangles.

D. S.A. 5 L.A. 1 B

E. S.A. 5 L.A. 1 2B

F. Uses height

G. Uses perimeter

H. Uses slant height

293 Lesson 11-3


pyramid slant height lateral area surface area cone

Rate how well you can fi nd the surface area of pyramids and cones.

Compare and Contrast How are the formulas for the surface area of a prism and the surface area of a pyramid alike? How are they diff erent?

30. Use the descriptions in the list at the right. Write the letter for each description under the correct polyhedron.

Prism Pyramid

31. How are the formulas alike? How are they different?

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________


Finding the Lateral Area of a Cone

Got It? What is the lateral area of a traffic cone with radius 10 in. and height 28 in.? Round to the nearest whole number.

26. Let / be the slant height of the cone. Label themissing dimensions on the cone at the right.

27. Use the Pythagorean Theorem to find /. 28. Use the formula for lateral area of a cone.

/ 5 Å 2 1 2 L.A. 5 p( )( )

5 Å 1 5 p( )( )

5 Å <

29. To the nearest square inch, the lateral area of the traffic cone is in.2.

10

100 784 10

934.0626473

934

!884

884

r28 l

Sample: Alike: Bases are polygons; both use perimeter.

Different: L.A. for pyramids uses triangular faces and slant

height and S.A. has only one base. L.A. for prisms uses

rectangular faces and height and S.A. has two bases.

A B E F G A C D G H

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Vocabulary

Review

Chapter 11 294

11-4 Volumes of Prisms and Cylinders

Label each diagram cylinder or prism.

1. 2. 3. 4.

Vocabulary Builder

composite (adjective, noun) kum PAHZ it

Related Words: compound, combination, component

Definition: Composite means put together with distinct parts.

Main Idea: A composite is a whole made up of different parts.

Use Your Vocabulary

Complete each statement with the correct phrase from the list below. Use each phrase only once.

composite function composite map composite number composite sketch

5. A 9 combines different descriptions of features.

6. A 9 has factors other than one and the number.

7. A 9 shows the locations of shopping malls, houses, and roads in one illustration.

8. A 9 shows how to apply at least one function to another function.

prismprismcylindercylinder

composite sketch

composite number

composite map

composite function

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Problem 1

Theorem 11-5 Cavalieri’s Principle

Theorem 11-6 Volume of a Prism

5 ft3 ft

4 ft

Bh

3 cm3 cm

3 cm

3 cm

3 cm 6 cm

295 Lesson 11-4

Finding the Volume of a Rectangular Prism

Got It? What is the volume of the rectangular prism at the right?

13. Circle the measurements of a base of the prism.

3 ft 3 4 ft 3 ft 3 5 ft 4 ft 3 5 ft

14. Underline the correct word to complete the sentence.

The base is a rectangle / square .

15. Find B. 16. Find V.

B 5 ? V 5 ?

5 ? 5 ?

5 5

17. Underline the correct word to complete the sentence.

The units for this volume are cubic / square feet .

18. The volume of the prism is .

If two space fi gures have the same height and the same cross-sectional area at every level, then they have the same volume.

9. The three prisms below have the same height and the same volume. The first is a square prism. Label the missing dimensions.

Th e volume of a prism is the product of the area of the base and the height of the prism.

V 5 Bh

11. A prism with a base area of 15 m2 and a height 4 m has a volume of m3.

12. A prism with a volume of 81 ft3 and a height of 3 ft has a base area of ft2.

10. Circle the solid(s) that may have the same cross-sectional area at every level.

cone cylinder prism pyramid

60

<

5

15

60 ft3

w

3

B

15

60

h

4

27

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Problem 3

Th e volume of a cylinder is the product of the area of the base and the height of the cylinder.

V 5 Bh, or V 5 pr2h

25. Shade a base of the cylinder at the right.

26. Describe the shape of the base.

_________________________________________________________

Theorem 11-7 Volume of a Cylinder

Problem 2

h

r

WriteThink

First I need to find the radius.

Now I simplify.

I can use the formula V r2h and

substitute for r and h.

21r

2

1 3

V

2

3

V

10 m

6 m5 m

2 m3 m

Chapter 11 296

Finding the Volume of a Triangular Prism

Got It? What is the volume of the triangular prism at the right?

19. The base is a right triangle with legs of length m and m.

20. The height of the prism is m.

21. Complete the formula for volume of a prism. V 5 ?

22. Find the area of the base. 23. Find the volume of the prism.

A 5 12 ? ? V 5 ?

5 12 ? 5

5

24. The volume of the triangular prism is m3.

Finding the Volume of a Cylinder

Got It? What is the volume of the cylinder at the right in terms of π?

27. Compete the reasoning model below.

28. The volume of the cylinder is m3.

10

10 30

B h

150

150

60

5

5

6

6

3π

The base is a circle.

30

Answers may vary. Sample:

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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Problem 4

10 in.6 in.

6 in.

3 in.

297 Lesson 11-4


Reasoning How is the volume of a rectangular prism with base 2 m by 3 m and height 4 m related to the volume of a rectangular prism with base 3 m by 4 m and height 2 m? Explain.

34. Cross out the formula that does NOT give the volume of a rectangular prism.

V 5 Bh V 5 pr2h V 5 /wh

35. The Commutative / Identity Property of Multiplication states that the product of factors is the same when listed in a different order.

36. Now answer the question.

______________________________________________________________________________________


volume composite space figure

Rate how well you can find the volume of prisms and cylinders.

Finding Volume of a Composite Figure

Got It? What is the approximate volume of the lunch box shown at the right? Round to the nearest cubic inch.

29. The top and bottom of the lunch box are sketched below. Label the dimensions.

30. Find the volume of the top. 31. Find the volume of the bottom. 32. Find the sum of the volumes.

V 5 12pr 2h V 5 Bh V 5 p 1

5 12 p(

2)( ) 5 ( ? )( ) <

5 5

33. The approximate volume of the lunch box is in.3.

10 in. 10 in.3 in.

6 in.

6 in.

3 10 6 6 10

45 360

501.3716694

36045π

501

Explanations may vary. Sample:

The volumes are equal by the Commutative Property of Multiplication.

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Vocabulary

Review

H

LW

Chapter 11 298

11-5 Volumes of Pyramids and Cones

1. Write L, W, or H to label the length, width, and height of the rectangular prism at the right.

2. Explain how the length, width, and height of a cube are related.

____________________________________________________________

Circle the correct statement in each exercise.

3. The width of a cylinder is the radius of a base of the cylinder.

The height of a cylinder is the length of an altitude of the cylinder.

4. The height of a pyramid is the length of a segment perpendicular to the base.

The slant height of a pyramid is the length of a segment perpendicular to the base.

Vocabulary Builder

volume (noun) VAHL yoom

Related Word: capacity

Main Idea: Volume measures quantity of space or amount, such as loudness of sound or a collection of books.

Definition: Volume is the amount of space that a three-dimensional figure occupies, measured in cubic units.

Example: The volume of a bottle of juice is 2 liters.

Use Your Vocabulary


5. A synonym for volume is capacity.

6. Volume is measured in square units.

7. You can find the volume of a circle.

The length, width and height of a cube are equal.

T

F

F

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Problem 1

Theorem 11-8 Volume of a Pyramid

Theorem 11-9 Volume of a Cone

Know Need Plan

B <

h 5

Volume of the pyramid Substitute the given values into the formula

.

B

h

B

h

299 Lesson 11-5

Finding Volume of a Pyramid

Got It? A sports arena shaped like a pyramid has a base area of about 300,000 ft2 and a height of 321 ft. What is the approximate volume of the arena?


13. Solve for V.

14. The approximate volume of the arena is ft3.

Th e volume of a pyramid is one third the product of the area of the base and the height of the pyramid.

8. Complete the formula for the volume of a pyramid.

V 5 ? Bh

Th e volume of a cone is one third the product of the areaof the base and the height of the cone.

V 5 13 Bh

9. Circle an equivalent formula for the volume of a cone.

V 5 13pr 2h V 5 1

3 ? 2pr ? h

Write the formula for the volume of each figure below.

10. 11.

V 5 13 ? ? V 5

13

300,000

321

32,100,000

V 5 13 Bh

N 13 (300,000)(321)

N 32,100,000

B h 13Bh

V 5 13Bh

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Problem 4

Problem 3

13 m

24 m

h

7 ft

6 ft

Problem 2

6 m

12 m

Chapter 11 300

Finding the Volume of a Pyramid

Got It? What is the volume of a square pyramid with base edges 24 m and slant height 13 m?

15. Label the pyramid at the right.

16. Find the height of 17. Find the area of 18. Find the volume of the pyramid. the base. the pyramid.

132 5 h2 1 2 B 5 ? V 5 ? Bh

169 5 h2 1 5 5 ? ?

h2 5 5

h 5

19. The volume of the pyramid is m3.

Finding the Volume of a Cone

Got It? A small child’s teepee is 6 ft high with a base diameter of 7 ft. What is the volume of the child’s teepee to the nearest cubic foot?

20. Label the cone at the right with the dimensions of the teepee.

21. The radius of the teepee is ft.

22. Use the justifications to find the volume of the teepee.

V 5 Use the formula with p for the volume of a cone.

V 5 Substitute for r and h.

5 Square the radius.

5 Simplify in terms of p.

< Use a calculator.

23. The volume of the child’s teepee to the nearest cubic foot is ft3.

Finding the Volume of an Oblique Cone

Got It? What is the volume of the oblique cone at the right in terms of p and rounded to the nearest cubic meter?

24. The radius of the base is m and the height is m.

25. Cross out the formula that is NOT a formula for the volume of a cone.

V 5 13 Bh V 5 Bh V 5 1

3pr 2h

12

144

25

5

960

3.5

77

6 12

13πr2h

13p(3.5)2(6)

13p(12.25)(6)

24.5p

76.96902001

960

576 13 576 5

24 2413

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0 2 4 6 8 10

Lesson Check

301 Lesson 11-5


pyramid cone oblique volume

Rate how well you can find the volumes of pyramids and cones.


Error Analysis A square pyramid has base edges 13 ft and height 10 ft. A cone has diameter 13 ft and height 10 ft. Your friend claims the fi gures have the same volume because the volume formulas for a pyramid and a cone are the same: V 5 1

3Bh. What is her error?

28. Is V 5 13 Bh the volume formula for both a pyramid and a cone? Yes / No


29. The base of a square pyramid is a circle / polygon .

30. The base of a cone is a circle / polygon .

31. Circle the base used in the formula for the volume of a cone. Underline the base used in the formula for the volume of a square pyramid.

B 5 pr 2 B 5 12bh B 5 s 2

32. Now explain your friend’s error.

_______________________________________________________________________

_______________________________________________________________________

26. Find the volume of the cone.

27. The volume of the cone in terms of p is m3.

Rounded to the nearest cubic meter, the volume of the cone is m3.

V 5 13 pr2h

5 13 p(6)2(12)

5 144p

N 452.3893421

144p

452

She forgot that the area formulas for the

bases of pyramids and cones are different.

Explanations may vary. Sample:

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5 cm

Chapter 11 302

11-6 Surface Areas and Volumes of Spheres


1. The diameter / radius of the circle at the right is 5 cm.

2. The circumference of a circle is the product of its diameter / radius and p.

3. The diameter / radius of a circle is a segment containing the center with

endpoints on the circle.

Vocabulary Builder

sphere (noun) sfeer

Related Words: spherical (adjective), hemisphere (noun)

Main Idea: A sphere is formed by the revolution of a circle about its diameter.

Definition: A sphere is the set of all points in space equidistant from a given point called the center.

Example: A basketball is a sphere.

Non-Example: A football is not a sphere.

Use Your Vocabulary

4. Complete each statement with sphere or spherical.

ADJECTIVE Each 9 candy looks like a rock.

NOUN A baseball is in the shape of a 9.


5. Celestial bodies such as the sun or Earth are often represented as spheres.

6. A sphere is a two-dimensional figure.

sphere

T

F

spherical

sphere

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Problem 1

Theorem 11-10 Surface Area of a Sphere

Know Need PlanThe circumference is

in. Th e radius r of the sphere

Th e surface area of the sphere

Solve the formula for circumference for .

Substitute into the formula for surface area of a sphere.

Problem 2

r

303 Lesson 11-6

Finding the Surface Area of a Sphere

Got It? What is the surface area of a sphere with a diameter of 14 in.? Give your answer in terms of π and rounded to the nearest square inch.

8. The radius of the sphere is in.

9. Find the surface area.

S.A. 5 4p( )2

Use the formula for surface area of a sphere.

5 4p( )2

Substitute for r.

5 p( ) Simplify.

< Use a calculator.

10. The surface area in terms of p is p in.2, or about in.2.

Finding Surface Area

Got It? What is the surface area of a melon with circumference 18 in.? Round your answer to the nearest ten square inches.


12. Find r in terms of p. 13. Use your value for r to find the surface area.

C 5 2pr S.A. 5 4pr2

14. To the nearest ten square inches, the surface area of the melon is in.2.

Th e surface area of a sphere is four times the product of p and the square of the radius of the sphere.

7. Complete: S.A. 5 ? ? 2

4

7

r

7

196

196

615.7521601

π r

18

100

S.A. 5 4πQ9π R2

5 4Q81π R

N 103.1324031

18 5 2πr 182π 5 r

r 5 9π

r

r

616

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Problem 4

Problem 3

The volumeof a sphere

Write

Relate four thirds the product of and thecube of the radius of the sphere.

V

is

43

r3

The diameter is 60 in., so

the radius is 30 in.43

r3The volume is .

Chapter 11 304

Finding the Volume of a Sphere

Got It? A sphere has a diameter of 60 in. What is its volume to the nearest cubic inch?

18. Complete the missing information in the diagram.

19. Complete to find the volume.

V 5 43p( )3

543p( )

5 p( )

<

20. The volume is about in.3.

Using Volume to Find Surface Area

Got It? The volume of a sphere is 4200 ft3. What is its surface area to the nearest tenth?

21. Circle the correct formula for the volume of a sphere.

V 5 43pr

2 V 5 43pr

3

15. Complete the model below.

Draw a line from each measure in Column A to its corresponding formula in Column B.

Column A Column B

16. surface area of a sphere 43pr

3

17. volume of a sphere 4pr2

Theorem 11-11 Volume of a Sphere

113,097

27,000

36,000

113,097.3355

30

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0 2 4 6 8 10

Lesson Check

WriteThink

I need to solve the volume formula for

the radius.

I can substitute the given volume into

the formula.

If I take the cube root of both sides, I

can solve for r. I need to use a

calculator to simplify.

Then, I can substitute r into the

formula for surface area of a sphere.

Finally, I can simplify.

Now, I can solve for r3.

10.00757003S.A. 4

1258.878021S.A.

43

r3V

4200 43

4200 43

r3

4200 r33

10.00891252 r

r 3

2

3(4200)

r3

4

4

305 Lesson 11-6


sphere (radius, diameter, circumference) great circle hemisphere

Rate how well you can find surface area and volume of a sphere.


Vocabulary What is the ratio of the area of a great circle to the surface area of the sphere?

24. A great circle is a circle whose center is the center of the 9.

25. A 5 26. S.A. 5 27. The ratio is , or .

22. Complete the reasoning model below.

23. To the nearest tenth of a foot, the surface area is ft2. 1258.9

πr2 1

44πr2

sphere

4πr2πr2

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Vocabulary

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Chapter 11 306

11-7 Areas and Volumes of Similar Solids

Complete each statement with the appropriate word from the list. Use each word only once.

similar similarities similarity similarly

1. There are several 9 house styles in the neighborhood.

2. The sequel may end 9 to the original movie.

3. There is not one 9 between his first and second drawings.

4. You could find several 9 between the two video games.

Vocabulary Builder

scale factor (noun) skayl FAK tur

Main Idea: The scale factor for similar polygons is the ratio of the lengths of their corresponding sides.

Definition: The scale factor for two similar solids is the ratio of their corresponding linear dimensions.

Example: The scale factor of a map may be 1 inch : 5 miles.

Use Your Vocabulary


5. You can find the scale factor for any two figures.

6. You can find the scale factor for similar figures.

7. You can write a scale factor as a ratio (a : b) or as a fraction QabR .

8. A scale factor is always written as the ratio a : b.

similar

similarly

similarity

similarities

F

T

T

F

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d.Problem 1

Theorem 11-12 Areas and Volumes of Similar Solids

Scale Factor ofSimilar Solids

a3 : b3a2 : b2

Ratio ofCorresponding

Areas

Ratio ofCorresponding

Volumes

a : b

6 12 105

307 Lesson 11-7

Identifying Similar Solids

Got It? Are the two cylinders similar? If so, what is the scale factor of the first figure to the second figure?


9. You can check whether prisms are similar by comparing the ratios of

two / three dimensions.

10. You can check whether cylinders are similar by comparing the ratios

of two / three dimensions.

11. Always compare the ratios of corresponding / similar dimensions.

12. The ratio of the height of the larger cylinder to the height of the smaller cylinder

is . Simplified, this ratio is .

13. The ratio of the radius of the larger cylinder to the radius of the smaller cylinder

is .

14. Underline the correct words to complete the sentence. The ratios you found in

Exercises 12 and 13 are/are not equal, so the cylinders are/are not similar.

15. The scale factor is .

If the scale factor of two similar solids is a : b, then

• the ratio of their corresponding areas is a2 : b2

• the ratio of their volumes is a3 : b3

16. Complete the model below.

17. If the scale factor of two similar solids is 1 : 2, then the ratio of their corresponding

areas is 2

: 2

, or : .

18. If the scale factor of two similar solids is 1 : 2, then the ratio of their corresponding

volumes is 3

: 3

, or : .

12

6

6

10

5

65

5

1 2 1 4

8121

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Problem 3

Problem 2

WriteThink

I can write the ratio of the lesser surface

area to the greater surface area.

I can take the square roots of the surface

areas and write the scale factor as a : b.

144

324

a2

b2

12

18

a

b

a2

b2

Chapter 11 308

Finding the Scale Factor

Got It? What is the scale factor of two similar prisms with surface areas 144 m2 and 324 m2?

19. Complete the reasoning model below.

20. The scale factor is : , which can be simplified to : .

Using a Scale Factor

Got It? The volumes of two similar solids are 128 m3 and 250 m3. The surface area of the larger solid is 250 m2. What is the surface area of the smaller solid?

21. Let ab represent the scale factor of the similar solids. Circle the simplified ratio of the volume of the smaller solid to the volume of the larger solid.

a3

b3 564

125 ab 5

64125 a3

b3 512564

22. Find the scale factor a : b. 23. Use the scale factor to find the ratio of surface areas.

ab 5

3%

3% 5 a2

b2 5

2

2 5

24. Circle the proportion that compares the surface area of the smaller solid to the surface area of the larger solid.

a2

b2 5x

250 ab 5

x250 a2

b2 5250

x

25. Now solve for x.

26. The surface area of the smaller solid is m2.

12 18 2 3

64 4 4 16

125

160

5 5 25

1625 5

x250

(250)(16) 5 25x

4000 5 25x

400025 5 25x

25 x 5 160

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0 2 4 6 8 10

Problem 4

a2

b24964

ab

78

The scale factor of thelarger cube to the smallercube is 7 : 8.

=

=

309 Lesson 11-7


Error Analysis Two cubes have surface areas 49 cm2 and 64 cm2. Your classmate tried to fi nd the scale factor of the larger cube to the smaller cube. Explain and correct your classmate’s error.

Underline the correct word(s) to complete each sentence.

30. The larger cube has the greater / lesser surface area.

31. The scale factor of larger cube to smaller cube is greater / less than the scale factor of smaller cube to larger cube.

32. Your classmate found the scale factor of the smaller / larger cube

to the smaller / larger cube.

33. Find the correct scale factor.

Begin with the proportion a2

b2 5 . Then, ab 5 .


similar solids scale factor

Rate how well you can find and use scale factors of similar solids.

Using a Scale Factor to Find Capacity

Got It? A marble paperweight shaped like a pyramid weighs 0.15 lb. How much does a similarly shaped marble paperweight weigh if each dimension is three times as large?

27. The scale factor of small paperweight to large paperweight is 1 : .

28. Let x 5 the weight of the larger paperweight. Use the justifications at the right to find the value of x.

13

35

0.15x The weights are proportional to their volumes.

1 50.15

x Simplify.

x 5 Use the Cross Products Property.

29. The paperweight that is three times as large weighs lb.

3

27

4.05

4.05

64

49

8

7

3

Documents

11-1 Space Figures and Cross Sections · Using Formulas to Find Surface Area of a Prism ... Complete the flow chart below. 15. The lateral area of the prism is m2. Finding Surface