Surface Areas and Volume of Spheres

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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

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
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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
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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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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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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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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
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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
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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
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