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CHAPTER 12 Extending Surface Area and Volume

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Learning Targets • Students will be able to draw isometric views of three-dimensional figures. • Students will be able to investigate cross-sections of three-dimensional figures.

Section 12.1 Notes: Representations of Three-dimensional Figures Vocabulary, Example

Types Definitions, Pictures and Examples

Cross Section

Types of Slice

From the above figures, Identify the type of shape resulting from a vertical,

angled and horizontal cross section of a cylinder.

Example 1:

Example 2:

Example 3:

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You Try!

Describe the shape resulting from a vertical, angled, and horiztonal cross section of a square pyramid. Vertical: Angled: Horizontal:

Examples of cross sections.

Example 1: BAKERY A customer ordered a two-layer sheet cake. Determine the shape of each cross section of the cake below.

Example 2 :Determine the shape of the cross section shown. a. b.

You Try!

Describe the shape resulting from each cross section.

a.

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b.

c.

Summary Describe the cross section.

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Learning Targets • Students will be able to find lateral areas and surface areas of prisms. • Students will be able to find lateral areas and surface areas of cylinders.

Section 12.2 Notes: Surface Areas of Prisms and Cylinders

Vocabulary, Example Types

Definitions, Pictures and Examples

Lateral Faces

Lateral Edges

Altitude

Base Edges

Height

Label the Parts

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Lateral Area

Lateral Area of a Prism

With a highlighter, Identify the base of the

given figures.

Example 1: a. b. c.

Example 2: The length of each side of the base of a regular octagonal prism is 6 inches, and the height is 11 inches. Find the lateral area.

Example 3: Lateral Area of a Prism Find the lateral area of the prism. Round your answers to the nearest hundredth.

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You Try!

Example 4: Find the lateral area of the prism.

Surface Area of a Prism

Surface Area of Prisms Example 5: Find the surface area of the rectangular prism Example 6: Find the surface area of a triangular prism. Round to the nearest tenth.

Example 7: Find the surface area of the regular hexagonal prism.

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You Try!

Example 8: Find the surface area of the regular pentagonal prism.

Lateral Area of a Right Cylinder

Surface area of a Cylinder

Example 9: Find the lateral area and the surface area of the cylinder. Round to the nearest thousandth. Example 10: Find the lateral area and the surface area of the cylinder. Round to the nearest tenth.

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You Try! Find the lateral area and surface area of each cylinder. Round to the nearest tenth. a. b.

Summary!

Find the lateral area and surface area of each prism. 1. 2.

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Learning Targets

• Students will be able to find lateral areas and surface areas of pyramids. • Students will be able to find lateral areas and surface areas of cones.

Section 12.3 Notes: Surface Areas of Pyramids and Cones

Vocabulary, Example

Types Definitions, Pictures and Examples

Regular Pyramid

Lateral Faces

Vertex

Lateral Edge

Slant Height

Label the Regular Pyramid with the given

information.

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Lateral Area of a Regular Pyramid

Examples in finding the lateral area of a pyramid.

Example 1: Find the lateral area of the square pyramid to the nearest tenth.

Surface Area of a Regular Pyramid

Examples of Surface area of pyramids

Example 2: Find the surface area of the square pyramid to the nearest tenth.

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Example 3: Find the surface area of the regular pyramid. Round to the nearest hundredth.

You Try! 1. Find the surface area of the square pyramid.

2. Find the surface area of the regular pyramid.

Lateral Area of a Cone

Find the lateral area of a cone

Example 4: The cone represents a conical slate roof on a house. Find the lateral area.

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Find the surface area of the cone.

Example 5: Find the surface area of the cone. Round to the nearest tenth.

You Try! 3. Find the surface area of the cone.

Using the solid and model provided, write in the formulas for each solid.

Solid Model Lateral Area Surface Area

Prism

Cylinder

Pyramid

Cone

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12.2-12.3 Extension: Surface Area of Composite Figures

Vocabulary, Example Types

Definitions, Pictures and Examples

Surface Area of a Composite Figure

Identify the solids that form the composite solid.

1.

2.

3.

Example 1: Identify the solids that make the composite solid and then find the surface area.

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Example 2: Identify the solids that make the composite solid and then find the surface area.

Example 3: Steven is making a wood box to fit two flower pots in the middle. His rough draft of the contraption looks like the diagram below. Find the amount of paint he needs to paint the outside and the inside holes.

Example 4: Identify the solids that make the composite figure. Find the total surface area of the composite space figures.

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Example 5: Identify the solids that make the composite figure. Find the surface area of the composite figure.

You Try!

1. Find the total surface area of the composite figure. 2. A storage bin is shaped as in the figure. The radius of the cylindrical top is 7 ft. The overall height of the bin is 26 ft. and the altitude of the conical section is 12 ft. Find the total surface area of the bin in bushels.

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3. The dimensions of the base of a rectangular solid are 5 and 8, and its altitude is 12. A hole, extending from upper base to lower base, is in the shape of a right triangular prism whose bases are equilateral ∆’s having an edge of 3. Determine the total surface area of the figure. 4. Jimmy’s lunch box in the shape of a half cylinder on a rectangular box. Find the total area of metal needed to manufacture it 5. A rectangular prism is 40 ft by 38 ft by 15 ft. Shown below is the prism with a half cylinder removed. Find the surface area of the original prism remaining.

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8

15 ft

38 ft

40 ft

10 ft 10 ft

12 ft

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Honors Geometry Name: _____________________________________ Sections 12.1 – 12.3 Quiz Review For all problems round to the nearest hundredth unless otherwise specified. 1. A cylinder has a lateral area of 120π square meters, and a height of 7 meters. Find the radius. 2. Find the surface area of the hexagonal prism. 3. Find the surface area of the prism. 4. Find the surface area of the solid figure at the right. 5. Use a right circular cone with a radius of 5 feet and a slant height of 12 feet. a) Find the lateral area. b) Find the surface area. 6. 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. 7. 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. 8. Find the surface area of the figure below.

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9. Find the surface area of the square pyramid below. 10. The following figure is a cylinder with a hole. Find the surface area of the figure. 11. The following figure is a cylinder with a 90° slice removed. Find the surface area of the figure. 12. The following figure consists of a prism with a half cylinder in top. Find the surface area of the figure. 13. Find the surface area of the figure below. 14. The total height of the tower shown is 10 m. If one liter of paint will cover an area of 10 m2, how many 1-L cans of paint are needed to pain the entire tower?

28 in.

50 in.

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5 3

12 m

6 m

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Learning Targets • Students will be able to find volumes of prisms. • Students will be able to find volumes of cylinders.

Section 12.4 Notes: Volumes of Prisms and Cylinders

Vocabulary, Example Types

Definitions, Pictures and Examples

Volume of a Prism

Example 1: Find the volume of the prism. Make sure to label the correct units.

Example 2: Find the volume of the prism. Make sure you label the correct units.

You Try! 1. Find the volume of the prism. Make sure you label the correct units.

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Volume Of a Cylinder

Example 2: Find the volume of the cylinder. Make sure to label the correct units and leave your answer in terms of π.

You Try! Find the volume of the cylinder.

Cavalieri’s Principle

Example of Cavalieri’s Principle

Example 3: Find the volume of the oblique cylinder. Label your units and leave your answer in terms of π.

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Summary! Find the volume of the given prism.

Find the volume of the cylinder.

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Learning Targets • Students will be able to find volumes of pyramids.

• Students will be able to find volumes of cones • Section 12.5 Notes: Volumes of Pyramids and Cones

Vocabulary, Example

Types Definitions, Pictures and Examples

Volume of a Pyramid

Example 1: Find the volume of the square pyramid. Label the units.

Example 2: Find the volume of the square pyramid. Label the units.

You Try! 1. Find the volume of the pentagonal pyramid. Label the units.

Volume of a Cone

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Example 2: Find the volume of the cone. Leave your answers in terms of π and label the units. Example 3: Find the volume of the oblique cone. Round your answers to the tenth.

You Try!

1. Find the volume of each cone.

In the below chart, fill out the volume formulas for each individual solid.

Solid Prism Cylinder Pyramid Cone

Model

Volume

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Learning Targets • Students will be able to find surface areas of spheres. • Students will be able to find volumes of spheres.

Section 12.6 Notes: Surface Areas and Volumes of Spheres

Vocabulary, Example Types

Definitions, Pictures and Examples

Surface Area of a Sphere

Find the surface area

Example 1: Find the surface area of the sphere. Round to the nearest tenth and label the units.

A plane can intersect a sphere in a point or in a circle. If the circle contains the center of the sphere, the

intersection is called a _______________________. The endpoints of a diameter of a great circle are called the _________.

Since a great circle has the same center as the sphere and its radii are also radii of the sphere, it is the largest circle that can be drawn on a sphere. A great circle separates a sphere into two congruent halves, called ________________.

Surface Area of a Hemisphere

Example 2: Find the surface area of the hemisphere. Label the units.

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Example 3: Find the surface area of the sphere if the circumference of the great circle is 10π. Example 4: Find the surface area of the sphere if the area of the great circle is approximately 160 square meters.

Volume of a Sphere

Example 5: Find the volume of the sphere. Round to the nearest hundredth and label the units. Example 6: Find the volume of the sphere with a great circle that has a circumference of 30π centimeters. Round to the nearest tenth.

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Volume of a Hemisphere

Example 7: Find the volume of the hemisphere with a diameter of 6 ft. Example 8 RECESS The jungle gym outside of Jada’s school is a perfect hemisphere. It has a volume of 4,000π cubic feet. What is the diameter of the jungle gym?

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12.4-12.6 Extension: Volume of Composite Figures

Vocabulary, Example Types

Definitions, Pictures and Examples

Volume of a Composite Figure

Example 1: Identify the solids that make the composite solid and then find the volume.

Example 2: Identify the solids that make the composite solid and then find the volume.

Example 3: Steven is making a wood box to fit two flower pots in the middle. His rough draft of the contraption looks like the diagram below. Find the amount of wood he needs to make the box.

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Example 4: Identify the solids that make the composite figure. Find the total volume of the composite space figures.

Example 5: Identify the solids that make the composite figure. Find the volume of the composite figure.

You Try!

1. Find the total volume of the composite figure.

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2. A storage bin is shaped as in the figure. The radius of the cylindrical top is 7 ft. The overall height of the bin is 26 ft. and the altitude of the conical section is 12 ft. Find the total volume of the bin in bushels. 3. The dimensions of the base of a rectangular solid are 5 and 8, and its altitude is 12. A hole, extending from upper base to lower base, is in the shape of a right triangular prism whose bases are equilateral ∆’s having an edge of 3. Determine the total volume of the figure. 4. Jimmy’s lunch box in the shape of a half cylinder on a rectangular box. Find the total volume of the lunch box.

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5. A rectangular prism is 40 ft by 38 ft by 15 ft. Shown below is the prism with a half cylinder removed. Find the volume of the original prism remaining.

15 ft

38 ft

40 ft

10 ft 10 ft

12 ft

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Honors Geometry Name: _____________________________________ Chapter 12 Review Sheet You may want to do your work on a separate sheet of paper! Round all answers to the nearest hundredth. 1. Find the total surface area of the box. The top is open. Use feet as the units. 2. Find the surface area of a sphere whose diameter is 7 cm. For numbers 3 – 5, refer to the diagram and given information: The height of the cone is 24 m and the lateral edge is 26 m. 3. Find the lateral area. 4. Find the surface area. 5. Find the volume. For numbers 6 – 8, refer to the diagram and given information: A right square-based pyramid with a base edge of 6 in. and slant height of 4 in. 6. Find the height of the pyramid. 7. Find the surface area of the pyramid. 8. Find the volume of the pyramid. For numbers 9 – 12, refer to the diagram and given information: A right prism, with equilateral bases. 9. Find the area of the base. 10. Find the volume. 11. Find the lateral area. 12. Find the surface area. For numbers 13 – 16, refer to the diagram and given information: A right circular cone. 13. Find the area of the base. 14. Find the volume. 15. Find the lateral area.

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16. Find the surface area. 17. Find the surface area of a sphere that has a radius of 6 mm. 18. Find the volume of a sphere that has a radius of 6 mm. 19. Find the volume of a cube with side length of 4 km. 20. F = (0, 0, 0) D = (3, 7, 4) a) What are the coordinates of A, G, and H? b) Find the volume. c) Find the surface area. 21. Find the volume of the sphere. 22. Find the area of one face of a cube whose volume is 512. For numbers 23 & 24, use the diagram. 23. Find the surface area of the prism. 24. Find the volume of the prism. For numbers 25 & 26, use the diagram. 25. Find the surface area of the figure. 26. Find the volume of the figure. 27. If the base (circle O) of the right circular cylinder shown has a circumference of 12π and OA = 10 m, find the lateral area.

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28. Circle P is the base of the right circular cone. If the circumference of the circle is 12π and m∠PAB = 60°, find the lateral area of the cone. For numbers 29 & 30, refer to the diagram. 29. Find the surface area of the shell, inside and out. Do not forget to include the top and bottom rims. 30. Find the surface area of the largest sphere that will fit inside the shell. 31. Find the surface area of the regular square pyramid shown. 32. Find the slant height of a regular cone with a volume of 3 7π and a circumference of 5π at the base. 33. In the regular square pyramid, PA = 17 cm, and slant height PE = 15 cm. What is the volume of the pyramid? 34. The volume of a sphere is equal to its surface area. What is its diameter? 35. The volume of a pyramid is 432 in.3. Its altitude is 12 inches. What is the area of the cross section 3 inches above the base? For numbers 36 & 37, refer to the diagram. 36. Given a regular pyramid as shown, find the lateral area of the pyramid. 37. Find the volume.

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38. You are at the store and want to purchase pizza sauce that comes in a cylindrical container, which is a better deal: A can that has a diameter of 10 cm, height of 20.5 cm and costs $3.49 or can that has a diameter of 7.5 cm, height of 11 cm and costs $1.09? 39. A cubic tank with 4.6 m edges is filled with water. How much water will be left in the tank if some is drained off to fill a cylindrical tank with a radius of 2.2 m and a height of 4.6 m? 40. How much cat food would fit into a can that has a height of 14.5 cm and a diameter of 9 cm? How much paper (in cm2) would you need to make the label? 41. A birthday gift is 55 cm, 40 cm wide, and 5 cm high. The sheet of paper you want to use to wrap it measures 75 cm by 100 cm, is the paper large enough to wrap the gift? 42. What happens to the surface area of a rectangular prism is all three of its dimensions are doubled? Tripled? 43. A square pyramid has a base with an area of 40 cm2 and a volume of 100 cm3. What is the height of the pyramid? 44. A painter uses a roller to paint a wall. The roller has a radius of 5 inches and a height of 13 inches. In one roll, what is the area of the wall that he will paint? What is the area that he will paint in 3 rolls? 45. The parking lot at the local ice cream store has 5 cylindrical posts in front of the store. These posts need to be painted bright yellow. The diameter of each post is 2 feet and each post is 4 feet high. Approximately how many square feet must be painted? (Hint: you do not paint the bottom of the post... it is stuck in the floor). 46. The lateral area for a cylinder is 144 in.2. The height of the cylinder is 6 in. What is the radius of the cylinder? 47. The total surface area of a triangular pyramid is 18 m2. The triangular base is an equilateral triangle with the base a 4 m and a height of 3 m. Find the slant length. 48. You have a rectangular prism with a total surface area of 82 cm2. The base is a rectangle measuring 7 cm by 2 cm. Find the height of the prism. 49. The lateral area of a cylinder is 94.2 cm2. The height is 6 cm. What is the radius? 50. You are painting a room that is 18 ft long, 14 ft wide and 8 ft high. Find the area of the four walls that you are going to paint. If the paint costs $6.50 a gallon and each gallon covers 128 ft2 of wall, how much will it cost to paint the room? 51. Tommy and Ethan each have a cylinder. Tommy’s cylinder has a diameter of 6 inches and a height of 8 inches. Ethan’s cylinder has a diameter of 8 inches and a height of 6 inches. Whose cylinder has a larger surface area?

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