Lesson 10-7 The Surface Area of a Sphere

Lesson

10-7

Lesson 10-7

You might be wondering why a lesson on surface area is at the end of a chapter on volume. The answer is simple. We derive the formula for the surface area of a sphere from the volume formula.

To calculate surface area, we created a net of the surface and found the area of each part of the net. This assumes that the surface can be unfolded into parts that are fl at. But no part of a sphere’s surface can be made to be fl at. Another strategy is needed.

Breaking the Sphere into Pyramids

Even though there is no simple net for a sphere, we can still derive the formula for the surface area. The idea is to consider a solid sphere as being made up of “almost pyramids.” Think of each pyramid as having its apex at the center and its base being a triangle covering a region as in the geodesic dome pictured above. One such “pyramid,” with height h and base area B, is drawn in the sphere at the right. The solid is not exactly a pyramid because its base is not exactly a polygon. Even so, when the “almost pyramid” is small, its volume is close to that of a pyramid, 1 __ 3 Bh. Because h equals r, the radius of the sphere, each “almost pyramid” has a volume of 1 __ 3 Br.

The Surface Area

of a Sphere

The Surface Area of a Sphere 633

A snowman is made up of three spheres: one 3' in diameter, one 2' in diameter and one 1' in diameter. What is the volume of snow used to make the snowman?

Mental Math

BIG IDEA From the formula for the volume of a sphere, a formula for the surface area of a sphere can be deduced.

C

h

area

The Biosphere of Montreal, Quebec, Canada

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634 Formulas for Volume

Chapter 10

Now break up the entire sphere into “almost pyramids” with base areas B1, B2, B3, B4, and so on. The sum of all the Bs is the surface area of the sphere. The volume V of the sphere is the sum of the volumes of all the “almost pyramids” with base area B1, B2, B3, B4, . . .

V = 1 __ 3 B1r + 1 __ 3 B2r + 1 __ 3 B3r + 1 __ 3 B4r + . . .

V = 1 __ 3 r(B1 + B2 + B3 + B4 + . . .)

V = 1 __ 3 r · S.A.

Finding the Formula

Now substitute 4 __ 3 πr3 for the volume V.

4 __ 3 πr3 = 1 __ 3 r · S.A.

To solve for the surface area, multiply both sides by 3.

4πr3 = r · S.A.

Now divide by r.

4πr2 = S.A.

Thus, the surface area of any sphere with radius r is 4πr2.

Sphere Surface Area Formula

The surface area of a sphere with radius r is 4πr2.

S.A. = 4πr2

This formula indicates that the surface area of a sphere is equal to 4 times the area of a great circle of the sphere.

Example 1

Earth is almost a sphere with a diameter of approximately 7900 miles. Find

the surface area of Earth.

Solution Because the diameter is 7900 miles, the radius is 3950 miles. Thus, S.A. = 4π(3950)2 ≈ 196,000,000 square miles.

QY1

B1

B2

C

B3

B4 B5

QY1

The area of the United States is about 3,700,000 mi2. About what percent of Earth’s surface is the United States?



Lesson 10-7

Example 2

Find the volume of a sphere with a surface area of 64π cm2.

Solution First fi nd the radius. 4πr2 = 64π

4πr2

____

4π

= 64π

___

4π

r2 = 16

r = 4 cm

Now substitute this radius into the volume formula. V = 4

_

3 πr3

= 4

_

3 π(4)3

= 4

_

3 π(64)

= 256

___

3 π cm3

Both the surface area and volume of a hemisphere are half that of a sphere.

Example 3

The top of the fuel tank shown at the right is used on aerospace fl ights. It is

a hemisphere with a 42-inch diameter. Find the lateral area and volume of

this hemisphere.

Solution Start with the formulas for the surface area and volume of a sphere. S.A. = ? and V = ? . Because a hemisphere is half of a sphere, divide the expressions by ? to get formulas for the lateral area and volume of a hemisphere. L.A. = ? and V = ? . Now substitute ? , the radius of the fuel tank hemisphere. L.A. = ? ≈ ? in2 and V = ? ≈ ? in3.

QY2

Questions

COVERING THE IDEAS

1. Fill in the Blank The surface area of a sphere is ? times the area of one of its great circles.

2. Calculate the surface area of a sphere with radius 5".

3. A sphere has a diameter of 12 cm. a. Calculate the exact surface area of this sphere. b. Approximate this surface area to the nearest tenth of a

square centimeter.

GUIDED

QY2

Write a formula for the surface area of a hemisphere (including the circular base) in terms of its radius.


636 Formulas for Volume

Chapter 10

4. Find the exact volume of a sphere with a surface area of 400π in2.

5. A sphere is created using 750 cubic centimeters of clay. What is the surface area of this sphere?

6. The round part of one soup ladle is a hemisphere and holds 8 ounces, or approximately 237 cubic centimeters of liquid. What is the surface area of the ladle, to the nearest square centimeter? Will the ladle fi t into a container with a diameter of 9 cm?

APPLYING THE MATHEMATICS

7. Find the formula for the surface area of a sphere in terms of its diameter d.

8. The equatorial diameter of Jupiter is approximately 11 times as great as the Earth’s diameter.

a. How do the surface areas of the planets compare? b. How do the volumes of the planets compare? c. How do your answers in Parts a and b relate to the number 11?

9. A sphere of radius r fi ts exactly into a cylinder, touching the cylinder at the top, bottom, and sides as shown in the fi gure at the right. How does the total surface area of the sphere compare to the lateral surface area of the cylinder?

10. Softballs and baseballs are sized by the circumference of a great circle of the balls. A baseball is 9 inches in circumference and softballs come in three sizes: 11 inches, 12 inches, and 16 inches.

a. Find the amount of material it would take to cover each of the balls (surface area).

b. Is there a relationship between the ratios of the radii of any two balls and their respective surface areas?

11. The fi gure at the right was created using a right cone, a right cylinder, and a hemisphere.

a. Find the total volume of the solid. b. Find the total surface area of the shape.

12. The radius of Earth is approximately 3950 miles, and 2 __ 3 of the surface is water. The radius of the Moon is approximately 1740 miles, and none is water. Which object has more land area, the Moon or Earth, and how do you know?

13. Write a formula for the surface area of a sphere in terms of C, the circumference of a great circle on the sphere.

r

r

4' 9'

6'



Lesson 10-7

REVIEW

14. The mean diameter of the Sun is 109 times the mean diameter of Earth. Then the volume of the Sun is how many times the volume of Earth? (Lesson 10-6)

15. A tent is in the shape of a right square pyramid. It is 5 feet high, and its base has sides of length 4 feet. What is the tent’s volume? (Lesson 10-4)

16. You can make the lateral surface of two different cylinders by rolling a 210-by-297 mm piece of paper along one or another of its sides.

a. What is the lateral area of each cylinder? (Lesson 9-9)

b. Which has more volume? (Lesson 10-3)

17. Explain why the two stacks of coins in the fi gure at the right have the same volume. (Lesson 10-3)

18. Consider two cylindrical jars of jam. One jar of jam is twice as tall as the other, but only half as wide. Which jar holds more jam? (Lesson 10-3)

19. Suppose solid A has volume 78 cm3 and solid B has volume 63 cm3. (Lesson 10-1)

a. Find the greatest possible volume of the union of the two solids. b. Find the least possible volume of the union of the two solids.

EXPLORATION

20. Another way to derive the formula for the surface area of a sphere from the volume formula is to imagine a sphere made of thin plastic. Let the plastic sphere be t units thick.

a. By subtracting the volume of the small sphere from the large sphere, fi nd an expression for the volume of the plastic sphere in terms of r and t.

b. Imagine that this plastic was laid out to form a cylindrical solid whose base area is the surface area of the sphere. The height of this cylindrical solid is t. Thus, the volume of this prism is V = S.A. · t. This expression and the expression in Part a are both expressions for the volume of the plastic, so they can be set equal to one another. Use this to fi nd an expression for S.A. in terms of r and t.

c. Now we let the thickness of the plastic become smaller and smaller until it is infi nitely small. What happens to the expression as t approaches 0?

r t

QY ANSWERS

1. about 2%

2. S.A. = 3πr2


Lesson 10-7 The Surface Area of a Sphere

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