1. Surface Area & Volume of Spheres
Section 9.6 C

2. Objectives:
Find the surface areas and volumes of spheres.

3. Key Vocabulary
Sphere
Hemisphere

4. Definitions
Sphere – the locus of points in space that are a given distance from a given point. (looks like a ball)
Center of a Sphere – the given point in the middle.
Radius of a Sphere – segment from the center to a point on the sphere.
Chord of a Sphere – a segment whose endpoints are on the sphere.
Diameter of a Sphere – a chord that goes through the center.

5. C is the center of the sphere.
Parts of a Sphere
C is the center of the sphere.
AB is a diameter.
CB & AC are radii.
DE & AB are chords. C A B E D

6. More Parts. . .
If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere.
Every great circle of a sphere separates a sphere into two congruent halves called hemispheres.
A hemisphere is half of a sphere.

7. Surface Area of a Sphere
Surface area of a sphere with radius r is S = 4r2.

8. The radius is 8 inches, so r = 8.
Example 1
Find the Surface Area of a Sphere
Find the surface area of the sphere. Round your answer to the nearest whole number. a. b.
SOLUTION a.
The radius is 8 inches, so r = 8.
S = 4πr2
= 4 · π · 82
≈ 804
The surface area is about 804 square inches. 8

9. The diameter is 10 cm, so the 10 2 = 5. So, r = 5. radius is
Example 1
Find the Surface Area of a Sphere b.
The diameter is 10 cm, so the 10 2
= 5. So, r = 5.
radius is
S = 4πr2
= 4 · π · 52
≈ 314
The surface area is about 314 square centimeters. 9

10. Your Turn:
Find the surface area of the sphere. Round your answer to the nearest whole number. 1.
ANSWER
201 in.2 2.
ANSWER
452 ft2 3.
ANSWER
616 in.2

11. Finding the Volume of a Sphere
Imagine that the interior of a sphere with radius r is approximated by n pyramids as shown, each with a base area of B and a height of r, as shown.
The volume of each pyramid is 1/3 Br and the sum is nB.

12. Finding the Volume of a Sphere
The surface area of the sphere is approximately equal to nB, or 4r2.
So, you can approximate the volume V of the sphere as follows:
V  n(1/3)Br
= 1/3 (nB)r
 1/3(4r2)r
=4/3r3
Each pyramid has a volume of 1/3Br.
Regroup factors.
Substitute 4r2 for nB.
Simplify.

13. Volume of a Sphere
Volume of a Sphere with radius r is V = 4/3r3.

14. The volume is about 34 cubic feet.
Example 2
Find the Volume of a Sphere
Find the volume of the sphere or hemisphere. Round your answer to the nearest whole number. a. b.
SOLUTION a.
V = 4 3
πr3
Write the formula for volume of a sphere. =
· π · 23 4 3
Substitute 2 for r. π = 32 3
Simplify. 23 = 2 · 2 · 2 = 8
≈ 34
Multiply.
ANSWER
The volume is about 34 cubic feet. 14

15. A hemisphere has half the volume of a sphere.
Example 2
Find the Surface Area of a Sphere
V = 1 2 4 3
πr3
Write the formula for the volume of a sphere. b.
A hemisphere has half the volume of a sphere.
Substitute 5 for r. = 1 2
· π · 53 4 3
Simplify. 53 = 5 · 5 · 5 = 125 π =
250 3
≈ 262
Multiply.
ANSWER
The volume is about 262 cubic inches. 15

16. The volume of air in the ball is about 905 cubic inches.
Example 3
Find the Volume of a Sphere
Estimate the volume of air in a beach ball that has a 12 inch diameter. Round your answer to the nearest whole number.
SOLUTION
V = 4 3
πr3
Write volume formula. =
· π · 63 4 3
Substitute for r. 12 2
= 6
= 288π
Simplify.
≈ 905
Multiply.
ANSWER
The volume of air in the ball is about 905 cubic inches. 16

17. Your Turn:
Find the volume to the nearest whole number. 1.
ANSWER
268 in.3 2.
ANSWER
57 cm3 3.
ANSWER
3054 ft3

18. Assignment
Pg , #1 – 49 odd