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7 mm

15 mm12 mm

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Math On the Spot

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L E S S O N

10.1Volume of Rectangular Prisms and Pyramids

ESSENTIAL QUESTIONHow do you find the volume of a rectangular prism and a rectangular pyramid?

Finding the Volume of a Rectangular PrismRemember that the volume of a rectangular prism is given by the formula

V = ℓ × w × h, or V = ℓwh.

The base of a rectangular prism is a rectangle with length ℓ and width w, so the

area of the base B is equal to ℓw. The volume formula can also be written as

V = Bh. In fact, the volume of any prism is the product of the base B and the

height h. Remember that volume is given in cubic units.

Find the volume of the rectangular prism.

Find the area of the base.

B = ℓw

B = 12 × 15

B = 180 m m 2

Find the volume.

V = Bh

V = 180 × 7

V = 1,260 m m 3

The volume of the rectangular prism is 1,260 cubic millimeters.

EXAMPLEXAMPLE 1

STEP 1

STEP 2

Equations, expressions, and relationships—7.8.A Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas. Also 7.9.A

7.9.A

Does it matter which dimension of a rectangle

you consider its height?

Volume of a Prism

The volume V of a prism is the area of its base B times its height h.

V = Bh

Use the formula.

Substitute for ℓ and w.

Use the formula.

Substitute for B and h.

315Lesson 10.1

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Net A Net B

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NetsA net is a two-dimensional pattern of shapes that can be folded into a

three-dimensional figure. The shapes in the net become the faces of the

three-dimensional figure.

Copy Net A and Net B on graph paper, and

cut them out along the blue lines.

One of these nets can be folded along the

black lines to make a cube. Which net will

not make a cube?

See if you can find another net that can be folded into a cube.

Draw a net that you think will make a cube on your graph paper, and

then cut it out. Can you fold it into a cube? Sketch your net below.

Compare your results with several of your classmates. How many

different nets for a cube did you and your classmates find?

STEP 1

STEP 2

STEP 3

EXPLORE ACTIVITY 1

Reflect1. What If? If you know the volume V and the height h of a prism,

how would you find the area of the base B?

2. Use the formula V = Bh to find the volume of a gift box

that is 3.5 inches high, 7 inches long, and 6 inches wide.

YOUR TURN

7.9.A

Unit 5316

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Height

Base

Exploring the Volume of a Rectangular PyramidA pyramid is a three-dimensional shape whose base is a polygon and whose other

faces are all triangles. Like a prism, a pyramid is named by the shape of its base.

Rectangular Pyramid Triangular Pyramid Pentagonal Pyramid

The faces of a pyramid that are not the base have a

common vertex, called the vertex of the pyramid.

The perpendicular distance from the vertex to the

base is the height of the pyramid.

Reflect3. What shapes will appear in a net for a rectangular prism that is not a

cube? How many of these shapes will there be?

How do you know that each net cannot be folded into a cube without

actually cutting and folding it?

4. 5.

6. Make a Conjecture If you draw a net for a cylinder, such as a soup can,

how many two-dimensional geometric shapes would this net have?

Name the shapes in the net for a cylinder.

Lesson 10.1 317

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In this activity, you will compare the volumes of a pyramid and a prism with

congruent bases and equal heights. Remember that congruent figures have

the same shape and size.

Make three-dimensional models. Make

larger versions of the nets shown. Make

sure the bases and heights in each net are

the same size. Fold each net, and tape it

together to form a prism or a pyramid.

Fill the pyramid with beans. Make sure that the beans are level

with the opening of the pyramid. Then pour the beans into the

prism. Repeat until the prism is full. How many times did you fill

the prism from the pyramid?

Write a fraction that compares the volume of the pyramid to the

volume of the prism.

volume of pyramid ________________

volume of prism = _____

Reflect7. Draw Conclusions A rectangular pyramid has a base area

of B and a height of h. What is a formula for the volume of

the pyramid? Justify your reasoning.

8. Communicate Mathematical Ideas The prism and the pyramid in this

activity have congruent bases and equal heights. Are they congruent

three-dimensional shapes? Explain.

STEP 1

STEP 2

STEP 3

EXPLORE ACTIVITY 2

9. The volume of a rectangular prism is 4 1 _ 2 in 3 . What is the volume of a

rectangular pyramid with a congruent base and the same height? Explain

your reasoning.

YOUR TURN

Math TalkMathematical Processes

7.8.A

Describe ways in which a prism and a pyramid

are different.

Unit 5318

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Math On the Spot

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Kyle needs to build a crate in the shape of a rectangular prism. The

crate must have a volume of 38 1_2

cubic feet, and a base area of 15 2_5

square feet. Find the height of the crate.

V = Bh

38 1 _ 2

= 15 2 _ 5

· h

77 __ 2

= 77 __ 5

h

5 __ 77

· 77 __ 2

= 77 __ 5

h · 5 __ 77

5 _ 2

= h

The height of the crate must be 5 _ 2

, or 2 1 _ 2

, feet.

A glass paperweight in the shape of a rectangular pyramid has a base

that is 4 inches by 3 inches and a height of 5 inches. Find the volume

of the paperweight.

V = 1_3

Bh

V = 1_3

· 12 · 5

V = 20

The paperweight has a volume of 20 cubic inches.

EXAMPLEXAMPLE 2

A

B

Solving Volume ProblemsYou can use the formulas for the volume of a rectangular prism and the volume

of a rectangular pyramid to solve problems.

10. A rectangular prism has a volume of 160 cubic centimeters

and a height of 4 centimeters. What is the area of its base?

11. A square pyramid has a base edge of 5.5 yards and a height of

3.25 yards. Find the volume of the pyramid to the nearest tenth.

YOUR TURN

7.9.A

Volume of a Rectangular Pyramid

The volume V of a pyramid is one-third the area of its base B times

its height h.

V = 1 _ 3

Bh

Use the formula.

Substitute for V and B.

Change the mixed numbers to fractions.

To divide both sides by 77 ___

5 , multiply both

sides by the reciprocal.

Use the formula.

Think: B = ℓw = 4 · 3 = 12

319Lesson 10.1

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9 ft

6 ft4 ft12

Guided Practice

1. Find the volume of the rectangular prism. (Example 1)

V = Bh

V = ( × _____ ) ( )

V = ft 3

Identify the three-dimensional shape that can be formed from each net.

(Explore Activity 1 and Explore Activity 2)

2. 3. 4.

5. The volume of a rectangular prism is

161.2 m 3 . The prism has a base that is

5.2 m by 3.1 m. Find the height of the

prism. (Example 2)

6. Explain how to use models to show the relationship between the volume of a rectangular prism

and a rectangular pyramid with congruent bases and heights.

ESSENTIAL QUESTION CHECK-IN??

V = Bh

=

( ×

) × h

=

h

= h

The height of the prism is .

Unit 5320

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48 in.16 in.

20 cm

12 cm10 cm

11 in.

4 in.

8 in.

9 in.

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Name Class Date

8. Explain the Error A student found the

volume of a rectangular pyramid with a

base area of 92 square meters and a height

of 54 meters to be 4,968 cubic meters.

Explain and correct the error.

9. A block of marble is in the shape of a

rectangular prism. The block is 3 feet long,

2 feet wide, and 18 inches high. What is the

volume of the block?

10. Multistep Curtis builds a doghouse with

base shaped like a cube and a roof shaped

like a pyramid. The cube has an edge

length of 3 1 _ 2 feet. The height of the pyramid

is 5 feet. Find the volume of the doghouse

rounded to the nearest tenth.

11. Miguel has an aquarium in the shape of a

rectangular prism. The base is 30.25 inches

long and 12.5 inches wide. The aquarium

is 12.75 inches high. What is the volume of

the aquarium to the nearest cubic inch?

12. After a snowfall, Sheree built a snow

pyramid. The pyramid had a square base

with side lengths of 32 inches and a height

of 28 inches. What was the volume of the

pyramid to the nearest cubic inch?

13. A storage chest has the shape of a

rectangular prism with the dimensions

shown. The volume of the storage chest is

18,432 cubic inches. What is its height?

14. Draw Conclusions A shipping company

ships certain boxes at a special rate. The

boxes must not have a volume greater

than 2,500 cm 3 . Can the box shown be

shipped at the special rate? Explain.

15. Communicate Mathematical Ideas Is the

figure shown a prism or a pyramid? Justify

your answer.

Independent Practice10.17.8.A, 7.9.A

321Lesson 10.1

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Work Area

16. A cereal box can hold 144 cubic inches of cereal. Suppose the box is

8 inches long and 1.5 inches wide. How tall is the box?

17. A shed in the shape of a rectangular prism has a volume of 1,080 cubic

feet. The height of the shed is 8 feet, and the width of its base is 9 feet.

What is the length of the shed?

18. Draw Conclusions Sue has a plastic paperweight shaped like a

rectangular pyramid. The volume is 120 cubic inches, the height is

6 inches, and the length is 10 inches. She has a gift box that is a

rectangular prism with a base that is 6 inches by 10 inches. How tall

must the box be for it to hold the pyramid?

19. Represent Real-World Problems A public swimming pool is in the

shape of a rectangular prism. The pool is 20 meters long and 16 meters

wide. The pool is filled to a depth of 1.75 meters.

a. Find the volume of water in the pool.

b. A cubic meter of water has a mass of 1,000 kilograms. Find the mass

of the water in the pool.

20. Analyze Relationships There are two glass pyramids at the Louvre

Museum in Paris, France. The outdoor pyramid has a square base with side

lengths of 35.4 meters and a height of 21.6 meters. The indoor pyramid has

a square base with side lengths of 15.5 meters and a height of 7 meters.

How many times as great is the volume of the outdoor pyramid than that

of the indoor pyramid?

21. Persevere in Problem Solving A small solid pyramid was installed

on top of the Washington Monument in 1884. The square base of the

pyramid is 13.9 centimeters on a side, and the height of the pyramid is

22.6 centimeters. The pyramid has a mass of 2.85 kilograms.

a. Find the volume of the pyramid. Round to the nearest hundredth.

b. Find the mass of the pyramid in grams.

c. Science The density of a substance is the ratio of its mass to its

volume. Find the density of the pyramid in grams per cubic

centimeter. Round to the nearest hundredth.

FOCUS ON HIGHER ORDER THINKING

Unit 5322

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