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The volume V of a pyramid is one-third the product of the area of the base and the height of the pyramid.

1.
English Spanish

7.3 Volumes of Pyramids

How can you find the volume of a pyramid?

S

STATE

STANDARDS

MA.7.G.2.1 1 ACTIVITY: Finding a Formula Experimentally

Work with a partner. 2 in. 2 in. 2 in.

● Draw the two nets on cardboard and cut

2 in.

them out.

2.25 in. 2 in.

2 in. 2 in. 2 in. 2 in.

2 in.

● Fold and tape the nets to form an open

square box and an open pyramid.

● Both figures should have the same size square base

and the same height.

● Fill the pyramid with pebbles. Then pour the pebbles

into the box. Repeat this until the box is full. How

many pyramids does it take to fill the box?

● Use your result to find a formula for the volume

of a pyramid.

2 ACTIVITY: Comparing Volumes

Work with a partner. You are an archeologist studying two ancient

pyramids. What factors would affect how long it took to build each

pyramid? Given similar conditions, which pyramid took longer to

build? Explain your reasoning.

Cholula Pyramid in Mexico Cheops Pyramid in Egypt

Height: about 217 ft Height: about 480 ft

Base: about 1476 ft by 1476 ft Base: about 755 ft by 755 ft

310 Chapter 7 Volumes of Solids

7.3 Volumes of Pyramids

How can you find the volume of a pyramid?

S

STATE

STANDARDS

MA.7.G.2.1 1 ACTIVITY: Finding a Formula Experimentally

Work with a partner. 2 in. 2 in. 2 in.

● Draw the two nets on cardboard and cut

2 in.

them out.

2.25 in. 2 in.

2 in. 2 in. 2 in. 2 in.

2 in.

● Fold and tape the nets to form an open

square box and an open pyramid.

● Both figures should have the same size square base

and the same height.

● Fill the pyramid with pebbles. Then pour the pebbles

into the box. Repeat this until the box is full. How

many pyramids does it take to fill the box?

● Use your result to find a formula for the volume

of a pyramid.

2 ACTIVITY: Comparing Volumes

Work with a partner. You are an archeologist studying two ancient

pyramids. What factors would affect how long it took to build each

pyramid? Given similar conditions, which pyramid took longer to

build? Explain your reasoning.

Cholula Pyramid in Mexico Cheops Pyramid in Egypt

Height: about 217 ft Height: about 480 ft

Base: about 1476 ft by 1476 ft Base: about 755 ft by 755 ft

310 Chapter 7 Volumes of Solids

2.
English Spanish

3 ACTIVITY: Finding and Using a Pattern

Work with a partner.

● Find the volumes of the pyramids.

● Organize your results in a table.

● Describe the pattern.

● Use your pattern to find the volume of a pyramid with a

side length and height of 20.

5

4

3

2

1

5

3 4

1 2

1 2 3 4 5

4 ACTIVITY: Breaking a Prism into Pyramids

Work with a partner. The rectangular prism can be cut to form three

pyramids. Show that the sum of the volumes of the three pyramids is

equal to the volume of the prism.

2

5 3

a. b. c.

5. IN YOUR OWN WORDS How can you find the volume of a pyramid?

6. Write a general formula for the volume of a pyramid.

Use what you learned about the volumes of pyramids to complete

Exercises 4 – 6 on page 314.

Section 7.3 Volumes of Pyramids 311

3 ACTIVITY: Finding and Using a Pattern

Work with a partner.

● Find the volumes of the pyramids.

● Organize your results in a table.

● Describe the pattern.

● Use your pattern to find the volume of a pyramid with a

side length and height of 20.

5

4

3

2

1

5

3 4

1 2

1 2 3 4 5

4 ACTIVITY: Breaking a Prism into Pyramids

Work with a partner. The rectangular prism can be cut to form three

pyramids. Show that the sum of the volumes of the three pyramids is

equal to the volume of the prism.

2

5 3

a. b. c.

5. IN YOUR OWN WORDS How can you find the volume of a pyramid?

6. Write a general formula for the volume of a pyramid.

Use what you learned about the volumes of pyramids to complete

Exercises 4 – 6 on page 314.

Section 7.3 Volumes of Pyramids 311

3.
English Spanish

7.3 Lesson

Lesson Tutorials

Volume of a Pyramid

Words The volume V of a pyramid is one-third

the product of the area of the base and height, h

the height of the pyramid.

Area of base

1 area of base, B

Algebra V = —Bh

3

Height of pyramid

EXAMPLE 1 Finding the Volume of a Pyramid

Find the volume of the pyramid.

1

V = —Bh Write formula for volume. 9 mm

3

1

= —(48)(9) Substitute.

3

B = 48 mm2

= 144 Multiply.

The volume is 144 cubic millimeters.

EXAMPLE 2 Finding the Volume of a Pyramid

Find the volume of the pyramid.

Study Tip a. b.

10 m

The area of the base of

a rectangular pyramid

7 ft

is the product of the

length ℓ and the 17.5 m

4 ft

width w. 3 ft 6m

1

You can use V = —ℓwh

3 1 1

to find the volume of a V = —Bh V = —Bh

rectangular pyramid. 3 3

1

= —(3)(4)(7)

3

1 1

()

= — — (17.5)(6)(10)

3 2

= 28 = 175

The volume is 28 cubic feet. The volume is 175 cubic

meters.

312 Chapter 7 Volumes of Solids

7.3 Lesson

Lesson Tutorials

Volume of a Pyramid

Words The volume V of a pyramid is one-third

the product of the area of the base and height, h

the height of the pyramid.

Area of base

1 area of base, B

Algebra V = —Bh

3

Height of pyramid

EXAMPLE 1 Finding the Volume of a Pyramid

Find the volume of the pyramid.

1

V = —Bh Write formula for volume. 9 mm

3

1

= —(48)(9) Substitute.

3

B = 48 mm2

= 144 Multiply.

The volume is 144 cubic millimeters.

EXAMPLE 2 Finding the Volume of a Pyramid

Find the volume of the pyramid.

Study Tip a. b.

10 m

The area of the base of

a rectangular pyramid

7 ft

is the product of the

length ℓ and the 17.5 m

4 ft

width w. 3 ft 6m

1

You can use V = —ℓwh

3 1 1

to find the volume of a V = —Bh V = —Bh

rectangular pyramid. 3 3

1

= —(3)(4)(7)

3

1 1

()

= — — (17.5)(6)(10)

3 2

= 28 = 175

The volume is 28 cubic feet. The volume is 175 cubic

meters.

312 Chapter 7 Volumes of Solids

4.
English Spanish

Find the volume of the pyramid.

Exercises 4–12 1. 2. 3.

7 in.

11 cm

6 ft

8 in.

10 in.

18 cm

7 cm

2

B =21 ft

EXAMPLE 3 Real-Life Application

(a) How many times more sunscreen is in Bottle B than in Bottle A?

(b) Which is the better buy?

Bottle A

a. Use the formula for the volume of a pyramid to estimate the

$9.96

amount of sunscreen in each bottle.

Bottle B

$14.40 Bottle A Bottle B

1 1

6 in. V = —Bh V = —Bh

3 3

4 in.

1 1

= —(2)(1)(6) = —(3)(1.5)(4)

3 3

1 in. 1.5 in.

2 in. 3 in. = 4 in.3 = 6 in.3

6

So, Bottle B has —, or 1.5 times more sunscreen than Bottle A.

4

b. Find the unit cost for each bottle.

Bottle A Bottle B

cost $9.96 cost $14.40

— = —3 —=—3

volume 4 in. volume 6 in.

$2.49 $2.40

= —3 = —3

1 in. 1 in.

The unit cost of Bottle B is less than the unit cost of Bottle A.

So, Bottle B is the better buy.

Bottle C

4. Bottle C is on sale for $13.20. Is Bottle C

Exercise 18 a better buy than Bottle B in Example 3? 3 in.

Explain.

2 in.

3 in.

Section 7.3 Volumes of Pyramids 313

Find the volume of the pyramid.

Exercises 4–12 1. 2. 3.

7 in.

11 cm

6 ft

8 in.

10 in.

18 cm

7 cm

2

B =21 ft

EXAMPLE 3 Real-Life Application

(a) How many times more sunscreen is in Bottle B than in Bottle A?

(b) Which is the better buy?

Bottle A

a. Use the formula for the volume of a pyramid to estimate the

$9.96

amount of sunscreen in each bottle.

Bottle B

$14.40 Bottle A Bottle B

1 1

6 in. V = —Bh V = —Bh

3 3

4 in.

1 1

= —(2)(1)(6) = —(3)(1.5)(4)

3 3

1 in. 1.5 in.

2 in. 3 in. = 4 in.3 = 6 in.3

6

So, Bottle B has —, or 1.5 times more sunscreen than Bottle A.

4

b. Find the unit cost for each bottle.

Bottle A Bottle B

cost $9.96 cost $14.40

— = —3 —=—3

volume 4 in. volume 6 in.

$2.49 $2.40

= —3 = —3

1 in. 1 in.

The unit cost of Bottle B is less than the unit cost of Bottle A.

So, Bottle B is the better buy.

Bottle C

4. Bottle C is on sale for $13.20. Is Bottle C

Exercise 18 a better buy than Bottle B in Example 3? 3 in.

Explain.

2 in.

3 in.

Section 7.3 Volumes of Pyramids 313

5.
English Spanish

7.3 Exercises

Help with Homework

1. WRITING How is the formula for the volume of a pyramid different from the

formula for the volume of a prism?

2. OPEN-ENDED Describe a real-life situation that involves finding the volume

of a pyramid.

3. REASONING A triangular pyramid and a triangular prism have the same

base and height. How many times greater is the volume of the prism than

the volume of the pyramid?

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Find the volume of the pyramid.

1 2 4. 5. 4 mm

6.

2 ft

8 yd

1 ft

2 ft B = 15 mm2

4 yd 5 yd

7. 8. 9.

8 in.

12 mm

7 cm

10 in. 6 in. 1 cm B = 63 mm2

3 cm

10. 11. 15 mm

7 ft

8 ft 14 mm

20 mm

6 ft

12. PARACHUTE In 1483, Leonardo da Vinci designed a

parachute. It is believed that this was the first parachute

ever designed. In a notebook, he wrote “If a man is

provided with a length of gummed linen cloth with a

length of 12 yards on each side and 12 yards high,

he can jump from any great height whatsoever

without injury.” Find the volume of air inside

Leonardo’s parachute.

Not drawn to scale

314 Chapter 7 Volumes of Solids

7.3 Exercises

Help with Homework

1. WRITING How is the formula for the volume of a pyramid different from the

formula for the volume of a prism?

2. OPEN-ENDED Describe a real-life situation that involves finding the volume

of a pyramid.

3. REASONING A triangular pyramid and a triangular prism have the same

base and height. How many times greater is the volume of the prism than

the volume of the pyramid?

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Find the volume of the pyramid.

1 2 4. 5. 4 mm

6.

2 ft

8 yd

1 ft

2 ft B = 15 mm2

4 yd 5 yd

7. 8. 9.

8 in.

12 mm

7 cm

10 in. 6 in. 1 cm B = 63 mm2

3 cm

10. 11. 15 mm

7 ft

8 ft 14 mm

20 mm

6 ft

12. PARACHUTE In 1483, Leonardo da Vinci designed a

parachute. It is believed that this was the first parachute

ever designed. In a notebook, he wrote “If a man is

provided with a length of gummed linen cloth with a

length of 12 yards on each side and 12 yards high,

he can jump from any great height whatsoever

without injury.” Find the volume of air inside

Leonardo’s parachute.

Not drawn to scale

314 Chapter 7 Volumes of Solids

6.
English Spanish

Copy and complete the table to find the area of the base B or the

height h of the pyramid.

Volume, V Area of Base, B Height,

ight, h

13. 60 in.3 6 in.

14. 144 cm3 48 cm2

15. 135 ft3 54 ft2

16. TEEPEE Use the photo of the teepee.

a. What is the shape of the base?

How can you tell?

ut

b. The teepee’s height is about

4 in.

me

10 feet. Estimate the volume

of the teepee.

17. PAPERWEIGHT How much h

3 in.

glass is needed to manufacture

3 in. 1000 paperweights? Explain your reasoning.

Paperweight

3 18. SPIRE Which sandcastle spire has a greater

volume? How much more sand is required to

6 in. 8 in. make the spire with the greater volume?

19. OPEN-ENDED A pyramid has a volume

B = 30 in.2 B = 24 in.2

of 40 cubic feet and a height of 6 feet.

Find one possible set of dimensions of

Spire A Spire B

the rectangular base.

3z

20. Do the two solids have the z

y y

same volume? Explain. x x

Simplify the expression. SKILLS REVIEW HANDBOOK

1 1 1 1

21. — × 12 × 7 22. — × 8 × 27 23. — × 62 × 5 24. — × 22 × 15

3 3 3 3

25. MULTIPLE CHOICE You spend 25% of your money on a shirt. Then you spend

1

— of the remainder on lunch. Lunch costs $8. What percent of your money is

6

spent on lunch? SECTION 4.1

○

A 4.2% ○

B 12.5% ○

C 16.7% ○

D 32%

Section 7.3 Volumes of Pyramids 315

Copy and complete the table to find the area of the base B or the

height h of the pyramid.

Volume, V Area of Base, B Height,

ight, h

13. 60 in.3 6 in.

14. 144 cm3 48 cm2

15. 135 ft3 54 ft2

16. TEEPEE Use the photo of the teepee.

a. What is the shape of the base?

How can you tell?

ut

b. The teepee’s height is about

4 in.

me

10 feet. Estimate the volume

of the teepee.

17. PAPERWEIGHT How much h

3 in.

glass is needed to manufacture

3 in. 1000 paperweights? Explain your reasoning.

Paperweight

3 18. SPIRE Which sandcastle spire has a greater

volume? How much more sand is required to

6 in. 8 in. make the spire with the greater volume?

19. OPEN-ENDED A pyramid has a volume

B = 30 in.2 B = 24 in.2

of 40 cubic feet and a height of 6 feet.

Find one possible set of dimensions of

Spire A Spire B

the rectangular base.

3z

20. Do the two solids have the z

y y

same volume? Explain. x x

Simplify the expression. SKILLS REVIEW HANDBOOK

1 1 1 1

21. — × 12 × 7 22. — × 8 × 27 23. — × 62 × 5 24. — × 22 × 15

3 3 3 3

25. MULTIPLE CHOICE You spend 25% of your money on a shirt. Then you spend

1

— of the remainder on lunch. Lunch costs $8. What percent of your money is

6

spent on lunch? SECTION 4.1

○

A 4.2% ○

B 12.5% ○

C 16.7% ○

D 32%

Section 7.3 Volumes of Pyramids 315