©Curriculum Associates, LLC Copying is not permitted. 261Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Name: Understand Volume of Cylinders, Cones, and Spheres

Study the example problem showing how to find the volume of a right prism. Then solve problems 1–7.

1 Why isn’t the base of this prism the 12 in. by 15 in. rectangular face?

2 If the bases of the case in the example were rectangles, what formula would you use to find B?

3 Suppose the area of the base of another triangular prism is 15 square inches and the height is 48 inches. What is the same about this prism and the prism in the example? What is different?

Example

A glass display case in the shape of a right triangular prism is shown at the right. What is the volume of the display case?

To find the volume of a right prism, you can use the formula V 5 Bh, where V is the volume, B is the area of the base, and h is the height of the prism.

Prerequisite: How can you find the volume of a right prism?

First find B, the area of the base. The bases of a triangular prism are triangles, so use the formula for the area of a triangle.

B 5 1 ·· 2 • 12 • 8

B 5 48

Now find the volume of the prism, where B 5 48 and h 5 15. V 5 Bh V 5 48 • 15 V 5 720

8 in.

15 in.

12 in.

The volume of the display case is 720 cubic inches.

Lesson 26

Vocabularyright prism a solid that

has two faces (called

bases) that are polygons.

The bases of a right

prism are the same size

and shape and are

parallel to each other.

The height of the prism

is the distance between

the bases.

base

height

base

right triangular prism

©Curriculum Associates, LLC Copying is not permitted.262 Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Solve.

Use the formula V 5 Bh for problems 4–7.

4 What is the volume of the right rectangular prism at the right?

5 A bakery box has the shape shown at the right.

a. Name the two types of prisms that are combined to make the shape of the bakery box.

b. Find the volume of the box.

Show your work.

Solution:

6 Which of the two popcorn containers has the greater volume? How much greater?

Show your work.

Solution:

7 The volume of the triangular prism at the right is 324 cubic centimeters. What is the height of the prism? Explain how you found your answer.

6 cm1.5 cm

2 cm

11 in.

12 in.

8 in.

8 in.

4 in. 3 in.

9 in.

5 in.

8 in.

6 in. 6 in.

A B

9 cm

15 cm

12 cm

? cm

©Curriculum Associates, LLC Copying is not permitted. 263Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Name: Lesson 26

Use Volume Formulas

Study the example problem showing how to use formulas to find the volume of a cone and a cylinder. Then solve problems 1–7.

1 What does pr2 in the example represent in the cylinder and in the cone?

2 If a cylinder has a volume of 75 cubic inches, what would be the volume of a cone with the same base and height? Explain.

3 If the height of the cylinder in the example were 12 centimeters, how would the volumes compare?

Example

The cone and cylinder at the right have the same radius and same height. Find and compare the volumes. Write the volumes in terms of p.

Volume of Cone

V 5 1 ·· 3 Bh

V 5 1 ·· 3 pr2h

V 5 1 ·· 3 • p • 22 • 6

V 5 8p cubic centimeters

Compare the volumes: 8p ···· 24p 5 1 ·· 3 , so the volume of the cone

is 1 ·· 3 of the volume of the cylinder.

6 cm6 cm

2 cm 2 cm

Volume of Cylinder

V 5 Bh

V 5 pr2h

V 5 p • 22 • 6

V 5 24p cubic centimeters

©Curriculum Associates, LLC Copying is not permitted.264 Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Solve.

4 The sphere and cylinder at the right have the same radius. Complete the equations to find the volumes of the sphere and the cylinder in terms of p. Then compare the volumes.

Volume of Sphere Volume of Cylinder

V 5 4 ·· 3 pr3 V 5 pr2h

V 5 4 ·· 3 p • V 5 p • •

V 5 4 ·· 3 p • V 5 p • •

V 5 cubic feet V 5 cubic feet

Volume of the sphere ················· Volume of cylinder 5 ———––– 5 ———–––

The volume of the sphere is times the volume of the cylinder.

5 Consider the cylinder in problem 4. If you double the length of the radius, what do you think will happen to the volume? Find the new volume in terms of p to check your prediction.

6 Suppose you double the radius and the height of the cylinder in problem 4. What do you think will happen to the volume? Find the new volume in terms of p to check your prediction.

7 The volume of a cylinder is 8p cubic centimeters. The radius and the height are equal. What is radius of the cylinder? Explain how you know.

6 ft

6 ft

6 ft

©Curriculum Associates, LLC Copying is not permitted. 265Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Name: Lesson 26

Reason and Write

Study the example. Underline two parts that you think make it a particularly good answer and a helpful example.

Example

Use the cone and cylinder shown below to complete the table. For both solid figures, compare the volume when the radius is halved to the original volume. Then compare the volume when the height is halved to the original volume. Describe the effects of halving the radius and halving the height.

3 in.

3 in.

3 in.

3 in.

Volume of a cone 5 1 ·· 3 pr2h

3 in.

3 in.

3 in.

3 in.

Volume of a cylinder 5 pr2h

Show your work. Use a table, numbers, and words to explain your answer.

Cone volume: radius halved: 1 ·· 3 p(1.5)2 (3) 5 2.25p

height halved: 1 ·· 3 p(3)2 (1.5) 5 4.5p

Cylinder volume: radius halved: p(1.5)2 (3) 5 6.75p

height halved: p(3)2 (1.5) 5 13.5p

Cone Volume (in.3) Cylinder Volume (in.3)

Original 9p 27p

Radius halved 2.25p 6.75p

Height halved 4.5p 13.5p

Cone: radius halved: 2.25p ····· 9p 5 1 ·· 4 ;

height halved: 4.5p ···· 9p 5 1 ·· 2

Cylinder: radius halved: 6.75p ····· 27p 5 1 ·· 4 ;

height halved: 13.5p ····· 27p 5 1 ·· 2

Halving the radius of a cone or cylinder results in a volume that is 1 ·· 4 of the original volume. Halving the height of a cone or cylinder results in a volume that is 1 ·· 2 of the original volume.

Where does the example . . . • answer all parts of

the problem?• use a table to

explain?• use numbers to

explain?• use words to

explain?

©Curriculum Associates, LLC Copying is not permitted.266 Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Solve the problem. Use what you learned from the model.

Use the cone and cylinder shown below to complete the table. For both solid figures, find the volume. Then compare the volume when the radius and height are both doubled to the original volume. Describe the effects of doubling both dimensions.

Volume of a cylinder 5 pr2h

4 ft

6 ft

4 ft

6 ft

Volume of a cone 5 1 ·· 3 pr2h

4 ft

6 ft

4 ft

6 ft

Show your work. Use a table, numbers, and words to explain your answer.

Cone Volume (ft3) Cylinder Volume (ft3)

Original

Radius and height doubled

Did you . . . • answer all parts of

the problem?• use a table to

explain?• use numbers to

explain?• use words to

explain?