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The volume of a cone is equal to one-third of the product area of circular base and height. The formula for the volume of a cone is (1/3)πr^2h, where, "h" is the height of the cone, and "r" is the radius of the base.

1.
English Spanish

7.4 Volumes of Cones

How can you remember the formulas for

S

STATE surface area and volume?

STANDARDS

MA.7.G.2.1

You discovered that the volume of a pyramid

is one-third the volume of a prism that has the

same base and same height. You can use a similar

activity to discover that the volume of a cone is Height â h

one-third the volume of a cylinder that has the

same base and height.

1

Volume of a Cone = —(Area of Base) × (Height)

3

Area of Base â B

1 ACTIVITY: Summarizing Volume Formulas

Work with a partner. You can remember the volume formulas for all of the

solids shown with just two concepts.

Volumes of Prisms and Cylinders

Volume = (Area of Base) × (Height)

Volumes of Pyramids and Cones

1

Volume = — (Volume of Prism or Cylinder with same base and height)

3

Make a list of all the formulas you need to remember to find the area of a base.

Talk about strategies for remembering these formulas.

Prism

Pyramid Cone

Cylinder

Prism Prism

316 Chapter 7 Volumes of Solids

7.4 Volumes of Cones

How can you remember the formulas for

S

STATE surface area and volume?

STANDARDS

MA.7.G.2.1

You discovered that the volume of a pyramid

is one-third the volume of a prism that has the

same base and same height. You can use a similar

activity to discover that the volume of a cone is Height â h

one-third the volume of a cylinder that has the

same base and height.

1

Volume of a Cone = —(Area of Base) × (Height)

3

Area of Base â B

1 ACTIVITY: Summarizing Volume Formulas

Work with a partner. You can remember the volume formulas for all of the

solids shown with just two concepts.

Volumes of Prisms and Cylinders

Volume = (Area of Base) × (Height)

Volumes of Pyramids and Cones

1

Volume = — (Volume of Prism or Cylinder with same base and height)

3

Make a list of all the formulas you need to remember to find the area of a base.

Talk about strategies for remembering these formulas.

Prism

Pyramid Cone

Cylinder

Prism Prism

316 Chapter 7 Volumes of Solids

2.
English Spanish

2 ACTIVITY: Volumes of Oblique Solids

Work with a partner. Think of a stack of paper. If you adjust the stack so that

the sides are oblique (slanted), do you change the volume of the stack? If the

volume of the stack does not change, then the formulas for volumes of right

solids also apply to oblique solids.

h=5 h=5

h=4 h=4

B = 4π B = 4π B = 9π B = 9π

Right cylinder Oblique cylinder Right cone Oblique cone

3 ACTIVITY: Summarizing Surface Area Formulas

Work with a partner. Make a list of the formulas for surface area that you

studied in Chapter 6. Organize these formulas in a way similar to what you

did in Activity 1.

Surface Area of a Right Prism =

Surface Area of a Right Pyramid =

Surface Area of a Right Cylinder =

Surface Area of a Right Cone =

4. IN YOUR OWN WORDS How can you remember the formulas for surface

area and volume? Write all of the surface area and volume formulas on a

summary sheet. Make the list short so that you do not have to memorize

many formulas.

Use what you learned about the volumes of cones to complete

Exercises 4– 6 on page 320.

Section 7.4 Volumes of Cones 317

2 ACTIVITY: Volumes of Oblique Solids

Work with a partner. Think of a stack of paper. If you adjust the stack so that

the sides are oblique (slanted), do you change the volume of the stack? If the

volume of the stack does not change, then the formulas for volumes of right

solids also apply to oblique solids.

h=5 h=5

h=4 h=4

B = 4π B = 4π B = 9π B = 9π

Right cylinder Oblique cylinder Right cone Oblique cone

3 ACTIVITY: Summarizing Surface Area Formulas

Work with a partner. Make a list of the formulas for surface area that you

studied in Chapter 6. Organize these formulas in a way similar to what you

did in Activity 1.

Surface Area of a Right Prism =

Surface Area of a Right Pyramid =

Surface Area of a Right Cylinder =

Surface Area of a Right Cone =

4. IN YOUR OWN WORDS How can you remember the formulas for surface

area and volume? Write all of the surface area and volume formulas on a

summary sheet. Make the list short so that you do not have to memorize

many formulas.

Use what you learned about the volumes of cones to complete

Exercises 4– 6 on page 320.

Section 7.4 Volumes of Cones 317

3.
English Spanish

7.4 Lesson

Lesson Tutorials

Volume of a Cone

height, h

Words The volume V of a cone is one-third

the product of the area of the base

and the height of the cone.

area of base, B

Area of base

1

Algebra V = —Bh

3

Height of cone

EXAMPLE 1 Finding the Volume of a Cone

Find the volume of the cone. Round your answer to the nearest tenth.

Study Tip The diameter is 4 meters. So, the radius is 2 meters.

Because B = π r 2, you 1

1 V = —Bh Write formula.

can use V = —π r 2h to 3

6m

3

find the volume of a 1

cone. = — π (2)2(6) Substitute.

3

= 8π ≈ 25.1 Simplify.

The volume is about 25.1 cubic meters. 4m

EXAMPLE 2 Finding the Height of a Cone

Find the height of the cone. Round your answer to the nearest tenth.

1

V = —Bh Write formula.

3 h

1

956 = — π (9)2(h) Substitute.

3

9 ft

956 = 27π h Simplify.

11.3 ≈ h Divide each side by 27π. Volume = 956 ft 3

The height is about 11.3 feet.

318 Chapter 7 Volumes of Solids

7.4 Lesson

Lesson Tutorials

Volume of a Cone

height, h

Words The volume V of a cone is one-third

the product of the area of the base

and the height of the cone.

area of base, B

Area of base

1

Algebra V = —Bh

3

Height of cone

EXAMPLE 1 Finding the Volume of a Cone

Find the volume of the cone. Round your answer to the nearest tenth.

Study Tip The diameter is 4 meters. So, the radius is 2 meters.

Because B = π r 2, you 1

1 V = —Bh Write formula.

can use V = —π r 2h to 3

6m

3

find the volume of a 1

cone. = — π (2)2(6) Substitute.

3

= 8π ≈ 25.1 Simplify.

The volume is about 25.1 cubic meters. 4m

EXAMPLE 2 Finding the Height of a Cone

Find the height of the cone. Round your answer to the nearest tenth.

1

V = —Bh Write formula.

3 h

1

956 = — π (9)2(h) Substitute.

3

9 ft

956 = 27π h Simplify.

11.3 ≈ h Divide each side by 27π. Volume = 956 ft 3

The height is about 11.3 feet.

318 Chapter 7 Volumes of Solids

4.
English Spanish

Find the volume V or height h of the cone. Round your answer to the

Exercises 4–17 nearest tenth.

1. 6 cm 2.

h≈

15 yd

15 cm

Volume = 7200 yd 3

V≈

EXAMPLE 3 Real-Life Application

30 mm You must answer a trivia question before the sand in the timer falls to

the bottom. The sand falls at a rate of 50 cubic millimeters per second.

10 mm How much time do you have to answer the question?

Use the formula for the volume of a cone to find the volume of the

sand in the timer.

24 mm 1

V = —Bh Write formula.

3

1

= — π (10)2(24) Substitute.

3

= 800π ≈ 2512 Simplify.

The volume of the sand is about 2512 cubic millimeters. To find the

amount of time you have to answer the question, multiply the volume

by the rate at which the sand falls.

1 sec

2512 mm3 × —3 = 50.24 sec

50 mm

You have about 50 seconds to answer the question.

3. WHAT IF? In Example 3, the sand falls at a rate of

60 cubic millimeters per second. How much time do

you have to answer the question?

4. WHAT IF? In Example 3, the height of the sand in the

timer is 12 millimeters and the radius is 5 millimeters.

How much time do you have to answer the question?

Section 7.4 Volumes of Cones 319

Find the volume V or height h of the cone. Round your answer to the

Exercises 4–17 nearest tenth.

1. 6 cm 2.

h≈

15 yd

15 cm

Volume = 7200 yd 3

V≈

EXAMPLE 3 Real-Life Application

30 mm You must answer a trivia question before the sand in the timer falls to

the bottom. The sand falls at a rate of 50 cubic millimeters per second.

10 mm How much time do you have to answer the question?

Use the formula for the volume of a cone to find the volume of the

sand in the timer.

24 mm 1

V = —Bh Write formula.

3

1

= — π (10)2(24) Substitute.

3

= 800π ≈ 2512 Simplify.

The volume of the sand is about 2512 cubic millimeters. To find the

amount of time you have to answer the question, multiply the volume

by the rate at which the sand falls.

1 sec

2512 mm3 × —3 = 50.24 sec

50 mm

You have about 50 seconds to answer the question.

3. WHAT IF? In Example 3, the sand falls at a rate of

60 cubic millimeters per second. How much time do

you have to answer the question?

4. WHAT IF? In Example 3, the height of the sand in the

timer is 12 millimeters and the radius is 5 millimeters.

How much time do you have to answer the question?

Section 7.4 Volumes of Cones 319

5.
English Spanish

7.4 Exercises

Help with Homework

1. VOCABULARY Describe the height of a cone.

2. WRITING Compare and contrast the formulas for the volume of a pyramid

and the volume of a cone.

3. REASONING You know the volume of a cylinder. How can you find the

volume of a cone with the same base and height?

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Find the volume of the cone. Round your answer to the nearest tenth.

1 4. 5. 6.

3m

4 in. 10 mm

6m

2 in. 5 mm

7. 2 ft 1 ft 8. 5 cm 9. 9 yd

6 yd

8 cm

10. 7 ft 11. 12. 4 cm

10 in.

3 ft

8 cm

5 in.

✗

13. ERROR ANALYSIS Describe and correct

1

the error in finding the volume of V = — Bh

8m 3

the cone.

1

= — (𝛑)(6)2(8)

3 cm 3

4 cm = 96𝛑 m3

6m

10 cm

8 cm

14. GLASS The inside of each glass is shaped

like a cone. Which glass can hold more liquid?

How much more?

320 Chapter 7 Volumes of Solids

7.4 Exercises

Help with Homework

1. VOCABULARY Describe the height of a cone.

2. WRITING Compare and contrast the formulas for the volume of a pyramid

and the volume of a cone.

3. REASONING You know the volume of a cylinder. How can you find the

volume of a cone with the same base and height?

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Find the volume of the cone. Round your answer to the nearest tenth.

1 4. 5. 6.

3m

4 in. 10 mm

6m

2 in. 5 mm

7. 2 ft 1 ft 8. 5 cm 9. 9 yd

6 yd

8 cm

10. 7 ft 11. 12. 4 cm

10 in.

3 ft

8 cm

5 in.

✗

13. ERROR ANALYSIS Describe and correct

1

the error in finding the volume of V = — Bh

8m 3

the cone.

1

= — (𝛑)(6)2(8)

3 cm 3

4 cm = 96𝛑 m3

6m

10 cm

8 cm

14. GLASS The inside of each glass is shaped

like a cone. Which glass can hold more liquid?

How much more?

320 Chapter 7 Volumes of Solids

6.
English Spanish

Find the height of the cone. Round your answer to the nearest tenth.

1

2 15. Volume = —π ft 3 16. Volume = 225 cm3 17. Volume = 3.6 in.3

18

1.8 in.

h

h h

2

3

10 cm

18. REASONING The volume of a cone is 20π cubic meters. What

4.8 in. is the volume of a cylinder having the same base and same

height?

19. VASE Water leaks from a crack in a vase at a rate of 0.5 cubic

inch per minute. How long does it take for 20% of the water to

10 in.

leak from a full vase?

20. LEMONADE STAND You have 10 gallons

of lemonade to sell. (1 gal ≈ 3785 cm3)

8 cm a. Each customer uses one paper cup.

How many paper cups will you need?

b. The cups are sold in packages of 50.

11 cm How many packages should you buy?

c. How many cups will be left over if

you sell 80% of the lemonade?

21. REASONING The cylinder and the cone have the same x ?

volume. What is the height of the cone?

y

22. Cone A has the same height but twice

2x

the radius of Cone B. What is the ratio of the volume

of Cone A to the volume of Cone B?

Find the volume of the solid. SECTION 7.1 SECTION 7.2 SECTION 7.3

23. 24. 25. 4 ft

9m 15 cm

9.5 ft

7m

5m 8 cm

10 cm

26. MULTIPLE CHOICE Which scale has a scale factor of 3 : 1? SECTION 5.4

○

A 1 in. : 2 ft ○

B 3 cm : 1 mm ○

C 5 ft : 15 yd ○

D 0.5 ft : 2 in.

Section 7.4 Volumes of Cones 321

Find the height of the cone. Round your answer to the nearest tenth.

1

2 15. Volume = —π ft 3 16. Volume = 225 cm3 17. Volume = 3.6 in.3

18

1.8 in.

h

h h

2

3

10 cm

18. REASONING The volume of a cone is 20π cubic meters. What

4.8 in. is the volume of a cylinder having the same base and same

height?

19. VASE Water leaks from a crack in a vase at a rate of 0.5 cubic

inch per minute. How long does it take for 20% of the water to

10 in.

leak from a full vase?

20. LEMONADE STAND You have 10 gallons

of lemonade to sell. (1 gal ≈ 3785 cm3)

8 cm a. Each customer uses one paper cup.

How many paper cups will you need?

b. The cups are sold in packages of 50.

11 cm How many packages should you buy?

c. How many cups will be left over if

you sell 80% of the lemonade?

21. REASONING The cylinder and the cone have the same x ?

volume. What is the height of the cone?

y

22. Cone A has the same height but twice

2x

the radius of Cone B. What is the ratio of the volume

of Cone A to the volume of Cone B?

Find the volume of the solid. SECTION 7.1 SECTION 7.2 SECTION 7.3

23. 24. 25. 4 ft

9m 15 cm

9.5 ft

7m

5m 8 cm

10 cm

26. MULTIPLE CHOICE Which scale has a scale factor of 3 : 1? SECTION 5.4

○

A 1 in. : 2 ft ○

B 3 cm : 1 mm ○

C 5 ft : 15 yd ○

D 0.5 ft : 2 in.

Section 7.4 Volumes of Cones 321