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This pdf includes the following topics:-

Surface Areas and Volumes of Spheres

Finding the Volume of a Sphere

Finding Surface Areas of Spheres

Finding the Diameter of a Sphere

Finding the Volume of a Composite Solid

Surface Areas and Volumes of Spheres

Finding the Volume of a Sphere

Finding Surface Areas of Spheres

Finding the Diameter of a Sphere

Finding the Volume of a Composite Solid

1.
11.8 Surface Areas and Volumes

of Spheres

Essential Question How can you find the surface area and the

volume of a sphere?

Finding the Surface Area of a Sphere

Work with a partner. Remove the covering from a baseball or softball.

r

USING TOOLS

STRATEGICALLY

To be proficient in math,

you need to identify You will end up with two “figure 8” pieces of material, as shown above. From the

relevant external amount of material it takes to cover the ball, what would you estimate the surface area

mathematical resources, S of the ball to be? Express your answer in terms of the radius r of the ball.

such as content located

on a website. S= Surface area of a sphere

Use the Internet or some other resource to confirm that the formula you wrote for the

surface area of a sphere is correct.

Finding the Volume of a Sphere

Work with a partner. A cylinder is circumscribed about a r

sphere, as shown. Write a formula for the volume V of the

cylinder in terms of the radius r.

r 2r

V= Volume of cylinder

When half of the sphere (a hemisphere) is filled with sand and

poured into the cylinder, it takes three hemispheres to fill the

cylinder. Use this information to write a formula for the volume V

of a sphere in terms of the radius r.

V= Volume of a sphere

Communicate Your Answer

3. How can you find the surface area and the volume of a sphere?

4. Use the results of Explorations 1 and 2 to find the surface area and the volume of

a sphere with a radius of (a) 3 inches and (b) 2 centimeters.

Section 11.8 Surface Areas and Volumes of Spheres 647

hs_geo_pe_1108.indd 647 1/19/15 3:31 PM

of Spheres

Essential Question How can you find the surface area and the

volume of a sphere?

Finding the Surface Area of a Sphere

Work with a partner. Remove the covering from a baseball or softball.

r

USING TOOLS

STRATEGICALLY

To be proficient in math,

you need to identify You will end up with two “figure 8” pieces of material, as shown above. From the

relevant external amount of material it takes to cover the ball, what would you estimate the surface area

mathematical resources, S of the ball to be? Express your answer in terms of the radius r of the ball.

such as content located

on a website. S= Surface area of a sphere

Use the Internet or some other resource to confirm that the formula you wrote for the

surface area of a sphere is correct.

Finding the Volume of a Sphere

Work with a partner. A cylinder is circumscribed about a r

sphere, as shown. Write a formula for the volume V of the

cylinder in terms of the radius r.

r 2r

V= Volume of cylinder

When half of the sphere (a hemisphere) is filled with sand and

poured into the cylinder, it takes three hemispheres to fill the

cylinder. Use this information to write a formula for the volume V

of a sphere in terms of the radius r.

V= Volume of a sphere

Communicate Your Answer

3. How can you find the surface area and the volume of a sphere?

4. Use the results of Explorations 1 and 2 to find the surface area and the volume of

a sphere with a radius of (a) 3 inches and (b) 2 centimeters.

Section 11.8 Surface Areas and Volumes of Spheres 647

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2.
11.8 Lesson What You Will Learn

Find surface areas of spheres.

Find volumes of spheres.

Core Vocabul

Vocabulary

larry

chord of a sphere, p. 648 Finding Surface Areas of Spheres

great circle, p. 648

A sphere is the set of all points in space equidistant from a given point. This point is

Previous called the center of the sphere. A radius of a sphere is a segment from the center to

sphere a point on the sphere. A chord of a sphere is a segment whose endpoints are on the

center of a sphere sphere. A diameter of a sphere is a chord that contains the center.

radius of a sphere

diameter of a sphere chord

hemisphere C

C radius diameter

center

As with circles, the terms radius and diameter also represent distances, and the

diameter is twice the radius.

If a plane intersects a sphere, then the

intersection is either a single point or a circle.

If the plane contains the center of the sphere, hemispheres

then the intersection is a great circle of the

great

sphere. The circumference of a great circle circle

is the circumference of the sphere. Every

great circle of a sphere separates the sphere

into two congruent halves called hemispheres.

Core Concept

Surface Area of a Sphere

The surface area S of a sphere is

r

S = 4πr 2

where r is the radius of the sphere. S = 4πr 2

To understand the formula for the surface

area of a sphere, think of a baseball. The

surface area of a baseball is sewn from

two congruent shapes, each of which

resembles two joined circles.

r

So, the entire covering of the baseball

consists of four circles, each with

radius r. The area A of a circle with

radius r is A = πr 2. So, the area of the leather covering

covering can be approximated by 4πr 2.

This is the formula for the surface area

of a sphere.

648 Chapter 11 Circumference, Area, and Volume

hs_geo_pe_1108.indd 648 1/19/15 3:31 PM

Find surface areas of spheres.

Find volumes of spheres.

Core Vocabul

Vocabulary

larry

chord of a sphere, p. 648 Finding Surface Areas of Spheres

great circle, p. 648

A sphere is the set of all points in space equidistant from a given point. This point is

Previous called the center of the sphere. A radius of a sphere is a segment from the center to

sphere a point on the sphere. A chord of a sphere is a segment whose endpoints are on the

center of a sphere sphere. A diameter of a sphere is a chord that contains the center.

radius of a sphere

diameter of a sphere chord

hemisphere C

C radius diameter

center

As with circles, the terms radius and diameter also represent distances, and the

diameter is twice the radius.

If a plane intersects a sphere, then the

intersection is either a single point or a circle.

If the plane contains the center of the sphere, hemispheres

then the intersection is a great circle of the

great

sphere. The circumference of a great circle circle

is the circumference of the sphere. Every

great circle of a sphere separates the sphere

into two congruent halves called hemispheres.

Core Concept

Surface Area of a Sphere

The surface area S of a sphere is

r

S = 4πr 2

where r is the radius of the sphere. S = 4πr 2

To understand the formula for the surface

area of a sphere, think of a baseball. The

surface area of a baseball is sewn from

two congruent shapes, each of which

resembles two joined circles.

r

So, the entire covering of the baseball

consists of four circles, each with

radius r. The area A of a circle with

radius r is A = πr 2. So, the area of the leather covering

covering can be approximated by 4πr 2.

This is the formula for the surface area

of a sphere.

648 Chapter 11 Circumference, Area, and Volume

hs_geo_pe_1108.indd 648 1/19/15 3:31 PM

3.
Finding the Surface Areas of Spheres

Find the surface area of each sphere.

a. b.

8 in. C = 12π ft

SOLUTION

a. S = 4πr2 Formula for surface area of a sphere

= 4π(8)2 Substitute 8 for r.

= 256π Simplify.

≈ 804.25 Use a calculator.

The surface area is 256π, or about 804.25 square inches.

12π

b. The circumference of the sphere is 12π, so the radius of the sphere is — = 6 feet.

2π

S = 4πr2 Formula for surface area of a sphere

= 4π(6)2 Substitute 6 for r.

= 144π Simplify.

≈ 452.39 Use a calculator.

The surface area is 144π, or about 452.39 square feet.

Finding the Diameter of a Sphere

Find the diameter of the sphere.

SOLUTION

S = 4πr2 Formula for surface area of a sphere

S = 20.25π cm2

20.25π = 4πr2 Substitute 20.25π for S.

COMMON ERROR

Be sure to multiply the 5.0625 = r2 Divide each side by 4π.

value of r by 2 to find 2.25 = r Find the positive square root.

the diameter.

The diameter is 2r = 2 • 2.25 = 4.5 centimeters.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Find the surface area of the sphere.

1. 40 ft 2.

C = 6π ft

3. Find the radius of the sphere.

S = 30π m2

Section 11.8 Surface Areas and Volumes of Spheres 649

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Find the surface area of each sphere.

a. b.

8 in. C = 12π ft

SOLUTION

a. S = 4πr2 Formula for surface area of a sphere

= 4π(8)2 Substitute 8 for r.

= 256π Simplify.

≈ 804.25 Use a calculator.

The surface area is 256π, or about 804.25 square inches.

12π

b. The circumference of the sphere is 12π, so the radius of the sphere is — = 6 feet.

2π

S = 4πr2 Formula for surface area of a sphere

= 4π(6)2 Substitute 6 for r.

= 144π Simplify.

≈ 452.39 Use a calculator.

The surface area is 144π, or about 452.39 square feet.

Finding the Diameter of a Sphere

Find the diameter of the sphere.

SOLUTION

S = 4πr2 Formula for surface area of a sphere

S = 20.25π cm2

20.25π = 4πr2 Substitute 20.25π for S.

COMMON ERROR

Be sure to multiply the 5.0625 = r2 Divide each side by 4π.

value of r by 2 to find 2.25 = r Find the positive square root.

the diameter.

The diameter is 2r = 2 • 2.25 = 4.5 centimeters.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Find the surface area of the sphere.

1. 40 ft 2.

C = 6π ft

3. Find the radius of the sphere.

S = 30π m2

Section 11.8 Surface Areas and Volumes of Spheres 649

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4.
Finding Volumes of Spheres

The figure shows a hemisphere and a cylinder with a cone removed. A plane parallel to

their bases intersects the solids z units above their bases.

r 2 − z2

r

z

r

r

Using the AA Similarity Theorem (Theorem 8.3), you can show that the radius of

the cross section of the cone at height z is z. The area of the cross section formed

by the plane is π(r 2 − z2) for both solids. Because the solids have the same height

and the same cross-sectional area at every level, they have the same volume by

Cavalieri’s Principle.

Vhemisphere = Vcylinder − Vcone

= πr 2(r) − —13 πr 2(r)

= —23 πr 3

So, the volume of a sphere of radius r is

⋅ ⋅

2 Vhemisphere = 2 —23 πr 3 = —43 πr 3.

Core Concept

Volume of a Sphere

The volume V of a sphere is

r

4

V = —π r 3

3

4

where r is the radius of the sphere. V = 3π r 3

Finding the Volume of a Sphere

Find the volume of the soccer ball. 4.5 in.

SOLUTION

V = —43 π r 3 Formula for volume of a sphere

= —43 π (4.5)3 Substitute 4.5 for r.

= 121.5π Simplify.

≈ 381.70 Use a calculator.

The volume of the soccer ball is 121.5π, or about 381.70 cubic inches.

650 Chapter 11 Circumference, Area, and Volume

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The figure shows a hemisphere and a cylinder with a cone removed. A plane parallel to

their bases intersects the solids z units above their bases.

r 2 − z2

r

z

r

r

Using the AA Similarity Theorem (Theorem 8.3), you can show that the radius of

the cross section of the cone at height z is z. The area of the cross section formed

by the plane is π(r 2 − z2) for both solids. Because the solids have the same height

and the same cross-sectional area at every level, they have the same volume by

Cavalieri’s Principle.

Vhemisphere = Vcylinder − Vcone

= πr 2(r) − —13 πr 2(r)

= —23 πr 3

So, the volume of a sphere of radius r is

⋅ ⋅

2 Vhemisphere = 2 —23 πr 3 = —43 πr 3.

Core Concept

Volume of a Sphere

The volume V of a sphere is

r

4

V = —π r 3

3

4

where r is the radius of the sphere. V = 3π r 3

Finding the Volume of a Sphere

Find the volume of the soccer ball. 4.5 in.

SOLUTION

V = —43 π r 3 Formula for volume of a sphere

= —43 π (4.5)3 Substitute 4.5 for r.

= 121.5π Simplify.

≈ 381.70 Use a calculator.

The volume of the soccer ball is 121.5π, or about 381.70 cubic inches.

650 Chapter 11 Circumference, Area, and Volume

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5.
Finding the Volume of a Sphere

The surface area of a sphere is 324π square centimeters. Find the volume of the sphere.

SOLUTION

Step 1 Use the surface area to find the radius.

S = 4πr2 Formula for surface area of a sphere

324π = 4πr2 Substitute 324π for S.

81 = r2 Divide each side by 4π.

9=r Find the positive square root.

The radius is 9 centimeters.

Step 2 Use the radius to find the volume.

V = —43 πr3 Formula for volume of a sphere

= —43 π (9)3 Substitute 9 for r.

= 972π Simplify.

≈ 3053.63 Use a calculator.

The volume is 972π, or about 3053.63 cubic centimeters.

Finding the Volume of a Composite Solid

Find the volume of the composite solid.

SOLUTION 2 in.

Volume Volume of Volume of 2 in.

= −

of solid cylinder hemisphere

(

= πr2h − —12 —43 πr3 ) Write formulas.

= π(2)2(2) − —23 π (2)3 Substitute.

16

= 8π − —3

π Multiply.

24

=— 16

π−— π Rewrite fractions using least

3 3 common denominator.

= —83π Subtract.

≈ 8.38 Use a calculator.

The volume is —83 π, or about 8.38 cubic inches.

1m Monitoring Progress Help in English and Spanish at BigIdeasMath.com

4. The radius of a sphere is 5 yards. Find the volume of the sphere.

5. The diameter of a sphere is 36 inches. Find the volume of the sphere.

5m

6. The surface area of a sphere is 576π square centimeters. Find the volume of

the sphere.

7. Find the volume of the composite solid at the left.

Section 11.8 Surface Areas and Volumes of Spheres 651

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The surface area of a sphere is 324π square centimeters. Find the volume of the sphere.

SOLUTION

Step 1 Use the surface area to find the radius.

S = 4πr2 Formula for surface area of a sphere

324π = 4πr2 Substitute 324π for S.

81 = r2 Divide each side by 4π.

9=r Find the positive square root.

The radius is 9 centimeters.

Step 2 Use the radius to find the volume.

V = —43 πr3 Formula for volume of a sphere

= —43 π (9)3 Substitute 9 for r.

= 972π Simplify.

≈ 3053.63 Use a calculator.

The volume is 972π, or about 3053.63 cubic centimeters.

Finding the Volume of a Composite Solid

Find the volume of the composite solid.

SOLUTION 2 in.

Volume Volume of Volume of 2 in.

= −

of solid cylinder hemisphere

(

= πr2h − —12 —43 πr3 ) Write formulas.

= π(2)2(2) − —23 π (2)3 Substitute.

16

= 8π − —3

π Multiply.

24

=— 16

π−— π Rewrite fractions using least

3 3 common denominator.

= —83π Subtract.

≈ 8.38 Use a calculator.

The volume is —83 π, or about 8.38 cubic inches.

1m Monitoring Progress Help in English and Spanish at BigIdeasMath.com

4. The radius of a sphere is 5 yards. Find the volume of the sphere.

5. The diameter of a sphere is 36 inches. Find the volume of the sphere.

5m

6. The surface area of a sphere is 576π square centimeters. Find the volume of

the sphere.

7. Find the volume of the composite solid at the left.

Section 11.8 Surface Areas and Volumes of Spheres 651

hs_geo_pe_1108.indd 651 1/19/15 3:31 PM

6.
11.8 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check

1. VOCABULARY When a plane intersects a sphere, what must be true for the intersection to be a

great circle?

2. WRITING Explain the difference between a sphere and a hemisphere.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, find the surface area of the sphere. In Exercises 13–18, find the volume of the sphere.

(See Example 1.) (See Example 3.)

3. 4. 13. 14.

7.5 cm

4 ft 8m 4 ft

5. 6. 15. 16.

22 yd 14 ft

18.3 m C = 4π ft

17. C = 20π cm 18. C = 7π in.

In Exercises 7–10, find the indicated measure.

(See Example 2.)

7. Find the radius of a sphere with a surface area of

4π square feet.

8. Find the radius of a sphere with a surface area of

1024π square inches. In Exercises 19 and 20, find the volume of the sphere

with the given surface area. (See Example 4.)

9. Find the diameter of a sphere with a surface area of

19. Surface area = 16π ft2

900π square meters.

20. Surface area = 484π cm2

10. Find the diameter of a sphere with a surface area of

196π square centimeters.

21. ERROR ANALYSIS Describe and correct the error in

finding the volume of the sphere.

In Exercises 11 and 12, find the surface area of the

✗

hemisphere.

11. 12. V = —43π (6)2

6 ft

5m 12 in. = 48π

≈ 150.80 ft3

652 Chapter 11 Circumference, Area, and Volume

hs_geo_pe_1108.indd 652 1/19/15 3:32 PM

Vocabulary and Core Concept Check

1. VOCABULARY When a plane intersects a sphere, what must be true for the intersection to be a

great circle?

2. WRITING Explain the difference between a sphere and a hemisphere.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, find the surface area of the sphere. In Exercises 13–18, find the volume of the sphere.

(See Example 1.) (See Example 3.)

3. 4. 13. 14.

7.5 cm

4 ft 8m 4 ft

5. 6. 15. 16.

22 yd 14 ft

18.3 m C = 4π ft

17. C = 20π cm 18. C = 7π in.

In Exercises 7–10, find the indicated measure.

(See Example 2.)

7. Find the radius of a sphere with a surface area of

4π square feet.

8. Find the radius of a sphere with a surface area of

1024π square inches. In Exercises 19 and 20, find the volume of the sphere

with the given surface area. (See Example 4.)

9. Find the diameter of a sphere with a surface area of

19. Surface area = 16π ft2

900π square meters.

20. Surface area = 484π cm2

10. Find the diameter of a sphere with a surface area of

196π square centimeters.

21. ERROR ANALYSIS Describe and correct the error in

finding the volume of the sphere.

In Exercises 11 and 12, find the surface area of the

✗

hemisphere.

11. 12. V = —43π (6)2

6 ft

5m 12 in. = 48π

≈ 150.80 ft3

652 Chapter 11 Circumference, Area, and Volume

hs_geo_pe_1108.indd 652 1/19/15 3:32 PM

7.
22. ERROR ANALYSIS Describe and correct the error in 33. MAKING AN ARGUMENT You friend claims that if

finding the volume of the sphere. the radius of a sphere is doubled, then the surface

area of the sphere will also be doubled. Is your friend

✗ 3 in.

V = —43 π (3)3

= 36π

correct? Explain your reasoning.

34. REASONING A semicircle with a diameter of

18 inches is rotated about its diameter. Find the

≈ 113.10 in.3 surface area and the volume of the solid formed.

35. MODELING WITH MATHEMATICS A silo has

the dimensions shown. The top of the silo is a

In Exercises 23–26, find the volume of the composite hemispherical shape. Find the volume of the silo.

solid. (See Example 5.)

23. 24.

6 ft

9 in.

60 ft

5 in. 12 ft

20 ft

25. 18 cm 26. 14 m

10 cm 6m

36. MODELING WITH MATHEMATICS Three tennis balls

are stored in a cylindrical container

with a height of 8 inches and a radius

of 1.43 inches. The circumference

In Exercises 27–32, find the surface area and volume of of a tennis ball is 8 inches.

the ball.

a. Find the volume of a tennis ball.

27. bowling ball 28. basketball

b. Find the amount of space within

the cylinder not taken up by the

tennis balls.

37. ANALYZING RELATIONSHIPS Use the table shown

for a sphere.

d = 8.5 in. C = 29.5 in. Radius Surface area Volume

3 in. 36π in.2 36π in.3

29. softball 30. golf ball 6 in.

9 in.

12 in.

C = 12 in. d = 1.7 in. a. Copy and complete the table. Leave your answers

in terms of π.

31. volleyball 32. baseball b. What happens to the surface area of the sphere

when the radius is doubled? tripled? quadrupled?

c. What happens to the volume of the sphere when

the radius is doubled? tripled? quadrupled?

38. MATHEMATICAL CONNECTIONS A sphere has a

C = 26 in. C = 9 in. diameter of 4(x + 3) centimeters and a surface area

of 784π square centimeters. Find the value of x.

Section 11.8 Surface Areas and Volumes of Spheres 653

hs_geo_pe_1108.indd 653 1/19/15 3:32 PM

finding the volume of the sphere. the radius of a sphere is doubled, then the surface

area of the sphere will also be doubled. Is your friend

✗ 3 in.

V = —43 π (3)3

= 36π

correct? Explain your reasoning.

34. REASONING A semicircle with a diameter of

18 inches is rotated about its diameter. Find the

≈ 113.10 in.3 surface area and the volume of the solid formed.

35. MODELING WITH MATHEMATICS A silo has

the dimensions shown. The top of the silo is a

In Exercises 23–26, find the volume of the composite hemispherical shape. Find the volume of the silo.

solid. (See Example 5.)

23. 24.

6 ft

9 in.

60 ft

5 in. 12 ft

20 ft

25. 18 cm 26. 14 m

10 cm 6m

36. MODELING WITH MATHEMATICS Three tennis balls

are stored in a cylindrical container

with a height of 8 inches and a radius

of 1.43 inches. The circumference

In Exercises 27–32, find the surface area and volume of of a tennis ball is 8 inches.

the ball.

a. Find the volume of a tennis ball.

27. bowling ball 28. basketball

b. Find the amount of space within

the cylinder not taken up by the

tennis balls.

37. ANALYZING RELATIONSHIPS Use the table shown

for a sphere.

d = 8.5 in. C = 29.5 in. Radius Surface area Volume

3 in. 36π in.2 36π in.3

29. softball 30. golf ball 6 in.

9 in.

12 in.

C = 12 in. d = 1.7 in. a. Copy and complete the table. Leave your answers

in terms of π.

31. volleyball 32. baseball b. What happens to the surface area of the sphere

when the radius is doubled? tripled? quadrupled?

c. What happens to the volume of the sphere when

the radius is doubled? tripled? quadrupled?

38. MATHEMATICAL CONNECTIONS A sphere has a

C = 26 in. C = 9 in. diameter of 4(x + 3) centimeters and a surface area

of 784π square centimeters. Find the value of x.

Section 11.8 Surface Areas and Volumes of Spheres 653

hs_geo_pe_1108.indd 653 1/19/15 3:32 PM

8.
39. MODELING WITH MATHEMATICS The radius of Earth 43. CRITICAL THINKING Let V be the volume of a sphere,

is about 3960 miles. The radius of the moon is about S be the surface area of the sphere, and r be the radius

1080 miles. of the sphere. Write an equation for V in terms of r

a. Find the surface area of Earth and the moon. ( V

and S. Hint: Start with the ratio —.

S )

b. Compare the surface areas of Earth and the moon.

c. About 70% of the surface of Earth is water. How 44. THOUGHT PROVOKING A spherical lune is the

many square miles of water are on Earth’s surface? region between two great circles of a sphere. Find

the formula for the area of a lune.

40. MODELING WITH MATHEMATICS The Torrid Zone

on Earth is the area between the Tropic of Cancer and

the Tropic of Capricorn. The distance between these

two tropics is about 3250 miles. You can estimate the

distance as the height of a cylindrical belt around the r

Earth at the equator. θ

Tropic of Cancer

Torrid

3250 mi

equator Zone 45. CRITICAL THINKING The volume of a right cylinder

is the same as the volume of a sphere. The radius of

the sphere is 1 inch. Give three possibilities for the

Tropic of

Capricorn dimensions of the cylinder.

a. Estimate the surface area of the Torrid Zone. 46. PROBLEM SOLVING A spherical cap is a portion of a

(The radius of Earth is about 3960 miles.) sphere cut off by a plane. The formula for the volume

πh

b. A meteorite is equally likely to hit anywhere on of a spherical cap is V = — (3a2 + h2), where a is

6

Earth. Estimate the probability that a meteorite the radius of the base of the cap and h is the height

will land in the Torrid Zone. of the cap. Use the diagram and given information to

find the volume of each spherical cap.

41. ABSTRACT REASONING A sphere is inscribed in a

a. r = 5 ft, a = 4 ft

cube with a volume of 64 cubic inches. What is the h

surface area of the sphere? Explain your reasoning. b. r = 34 cm, a = 30 cm a

r

c. r = 13 m, h = 8 m

42. HOW DO YOU SEE IT? The formula for the volume d. r = 75 in., h = 54 in.

of a hemisphere and a cone are shown. If each solid

has the same radius and r = h, which solid will have

a greater volume? Explain your reasoning.

47. CRITICAL THINKING A sphere with a radius of

r r 2 inches is inscribed in a right cone with a height

of 6 inches. Find the surface area and the volume

h of the cone.

2 1

V = 3π r 3 V = 3π r 2h

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Solve the triangle. Round decimal answers to the nearest tenth. (Section 9.7)

48. A = 26°, C = 35°, b = 13 49. B = 102°, C = 43°, b = 21

50. a = 23, b = 24, c = 20 51. A = 103°, b = 15, c = 24

654 Chapter 11 Circumference, Area, and Volume

hs_geo_pe_1108.indd 654 3/9/16 9:46 AM

is about 3960 miles. The radius of the moon is about S be the surface area of the sphere, and r be the radius

1080 miles. of the sphere. Write an equation for V in terms of r

a. Find the surface area of Earth and the moon. ( V

and S. Hint: Start with the ratio —.

S )

b. Compare the surface areas of Earth and the moon.

c. About 70% of the surface of Earth is water. How 44. THOUGHT PROVOKING A spherical lune is the

many square miles of water are on Earth’s surface? region between two great circles of a sphere. Find

the formula for the area of a lune.

40. MODELING WITH MATHEMATICS The Torrid Zone

on Earth is the area between the Tropic of Cancer and

the Tropic of Capricorn. The distance between these

two tropics is about 3250 miles. You can estimate the

distance as the height of a cylindrical belt around the r

Earth at the equator. θ

Tropic of Cancer

Torrid

3250 mi

equator Zone 45. CRITICAL THINKING The volume of a right cylinder

is the same as the volume of a sphere. The radius of

the sphere is 1 inch. Give three possibilities for the

Tropic of

Capricorn dimensions of the cylinder.

a. Estimate the surface area of the Torrid Zone. 46. PROBLEM SOLVING A spherical cap is a portion of a

(The radius of Earth is about 3960 miles.) sphere cut off by a plane. The formula for the volume

πh

b. A meteorite is equally likely to hit anywhere on of a spherical cap is V = — (3a2 + h2), where a is

6

Earth. Estimate the probability that a meteorite the radius of the base of the cap and h is the height

will land in the Torrid Zone. of the cap. Use the diagram and given information to

find the volume of each spherical cap.

41. ABSTRACT REASONING A sphere is inscribed in a

a. r = 5 ft, a = 4 ft

cube with a volume of 64 cubic inches. What is the h

surface area of the sphere? Explain your reasoning. b. r = 34 cm, a = 30 cm a

r

c. r = 13 m, h = 8 m

42. HOW DO YOU SEE IT? The formula for the volume d. r = 75 in., h = 54 in.

of a hemisphere and a cone are shown. If each solid

has the same radius and r = h, which solid will have

a greater volume? Explain your reasoning.

47. CRITICAL THINKING A sphere with a radius of

r r 2 inches is inscribed in a right cone with a height

of 6 inches. Find the surface area and the volume

h of the cone.

2 1

V = 3π r 3 V = 3π r 2h

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Solve the triangle. Round decimal answers to the nearest tenth. (Section 9.7)

48. A = 26°, C = 35°, b = 13 49. B = 102°, C = 43°, b = 21

50. a = 23, b = 24, c = 20 51. A = 103°, b = 15, c = 24

654 Chapter 11 Circumference, Area, and Volume

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