Surface Area and Volume of Spheres

Section 4.9

Standard: MM2G4 ab

Essential Questions: How do I find the surface area and volume of spheres? How does changing the radius or diameter affect the surface area and volume?

Formula: Surface Area of a Sphere

24S r

1. Find the surface area of the sphere, given that the radius is 9 inches.

24S rS = 4(9)2

S = 324S ≈ 1017.88 in.2

2. Find the surface area of the sphere, given that the diameter is 10 inches.

24S rS = 4(5)2

S = 100S ≈ 314.16 in.2

d = 2r10 = 2r 5 = r

3. Find the surface area of the sphere, given that the circumference of the sphere is ft.8

24S rS = 4(4)2

S = 64S ≈ 201.06 ft2 C = 2r

8 = 2r 4 = r

Formula: Volume of a Sphere

34

3V r

4. Find the volume of the sphere, given that the radius is 8 inches.

34

3V r

34(8)

3V

2048

3V

V ≈ 2144.66 in.3

5. Find the volume of the sphere, given that the diameter is 10 inches.

34

3V r

34(5)

3V

500

3V

V ≈ 523.60 in.3

d = 2r10 = 2r 5 = r

6. Find the volume of the sphere, given that the circumference of the sphere is ft.8

C = 2r8 = 2r 4 = r

34

3V r

34(4)

3V

256

3V

V ≈ 268.08 ft3

Complete the table.

Sphere 1 Sphere 2 Ratio of each for Sphere 1: Sphere 2

Radius 2 3 2 : 3Diameter

Circumference

Area of Great Circle

Surface Area

Volume

4 6 4:6 or 2:34 6 4:6 or 2:3

4 9 4:9 or 4:9

16 36 16:36 or 4:9

32

3 36

32 108:

3 38: 27or

Complete the table.

Sphere 1 Sphere 2 Ratio of each for Sphere 1: Sphere 2

Radius a b a : bDiameter

Circumference

Area of Great Circle

Surface Area

Volume

2a 2b a : b2a 2b a : b

a2 b2 a2 : b2

4a2 4b2 a2 : b2

34

3a 34

3b a3 : b3

Scale Factor _____________

Area Ratios _______________

Volume Ratios _______________

a : b

a2 : b2

a3 : b3

7. Two spheres have diameters 24 and 36. a. What is the ratio of the areas?

b. What is the ratio of the volumes?

Ratio of volumes:

Radii:Ratio of radii:

Ratio of areas:

24 and 3624 : 36 or 2 : 3

22 : 32 or 4 : 9

23 : 33 or 8 : 27

8. A sphere has a radius of 6 meters. The radius of a second sphere is 3 meters. (a.) How does the surface area of the second sphere compare the to surface area of the first sphere? (b.) Volumes?

Ratio of radii: 3 : 6

(a). Ratio of areas: 12 : 22

or 1 : 2

or 1 : 4

The second sphere is ¼ the size of the first. (The area of the second sphere is 4 times smaller)

(b). Ratio of volumes: 13 : 23 or 1 : 8

The second sphere is 1/8 the size of the first. (The volume of second sphere is 8 times smaller)

9. The radius of a sphere is 2.4 cm. (a.) How will the surface area change if the radius is doubled? (b.) Volume?

Double the radius: 2(2.4) = 4.8

Ratio of radii: 2.4 : 4.8 or 1 : 2

Ratio of surface areas: 12 : 22 or 1 : 4

Ratio of volumes: 13 : 23 or 1 : 8

The surface area is 4 times larger if the radius is doubled.

The volume is 8 times larger if the radius is doubled.

10. Find the surface area of one hemisphere of a circle if the circumference of a great

circle of the sphere is 7 cm.

C = 2r7 = 2r7/2 = r

Surface area of entire sphere: S = 4r2

S = 4 (3.5)2

S = 49S ≈ 153.93 cm2

Area of base of hemisphere: A = r2 A = (3.5)2

A = 12.25A ≈ 38.48 cm2

Surface area of hemisphere ≈ 153.93 cm2 + 38.48 cm2

≈ 192.41 cm2