This document provides information about calculating the surface area and volume of spheres. It begins with examples of finding the volume and surface area of spheres using the formulas V=4/3πr^3 and S=4πr^2. It then discusses how changing the radius affects volume and surface area. The document concludes with examples of finding the surface area and volume of composite figures made of spheres, cylinders, and hemispheres.

1. UNIT 11.6 SURFACE AREAUNIT 11.6 SURFACE AREA
AND VOLUME OF SPHERESAND VOLUME OF SPHERES

2. Warm Up
Find each measurement.
1. the radius of circle M if the diameter is 25 cm
2. the circumference of circle X if the radius is
42.5 in.
3. the area of circle T if the diameter is 26 ft
4. the circumference of circle N if the area is
625π cm2
12.5 cm
85π in.
169π ft2
50π cm

3. Learn and apply the formula for the
volume of a sphere.
Learn and apply the formula for the
surface area of a sphere.
Objectives

4. sphere
center of a sphere
radius of a sphere
hemisphere
great circle
Vocabulary

5. A sphere is the locus of points in space that are a
fixed distance from a given point called the center
of a sphere. A radius of a sphere connects the
center of the sphere to any point on the sphere. A
hemisphere is half of a sphere. A great circle
divides a sphere into two hemispheres

6. The figure shows a hemisphere and a cylinder with a
cone removed from its interior. The cross sections
have the same area at every level, so the volumes
are equal by Cavalieri’s Principle. You will prove that
the cross sections have equal areas in Exercise 39.
The height of the hemisphere is equal to the radius.

7. V(hemisphere) = V(cylinder) – V(cone)
The volume of a sphere with radius r is twice the
volume of the hemisphere, or .

9. Example 1A: Finding Volumes of Spheres
Find the volume of the sphere.
Give your answer in terms of π.
= 2304π in3
Simplify.
Volume of a sphere.

10. Example 1B: Finding Volumes of Spheres
Find the diameter of a sphere with volume
36,000π cm3
.
Substitute 36,000π for V.
27,000 = r3
r = 30
d = 60 cm d = 2r
Take the cube root of both sides.
Volume of a sphere.

11. Example 1C: Finding Volumes of Spheres
Find the volume of the hemisphere.
Volume of a hemisphere
Substitute 15 for r.
= 2250π m3
Simplify.

12. Check It Out! Example 1
Find the radius of a sphere with volume 2304π ft3
.
Volume of a sphere
Substitute for V.
r = 12 ft Simplify.

13. Example 2: Sports Application
A sporting goods store sells exercise balls in two
sizes, standard (22-in. diameter) and jumbo (34-
in. diameter). How many times as great is the
volume of a jumbo ball as the volume of a
standard ball?
standard ball:
jumbo ball:
A jumbo ball is about 3.7 times as great in volume
as a standard ball.

14. Check It Out! Example 2
A hummingbird eyeball has a diameter of
approximately 0.6 cm. How many times as great
is the volume of a human eyeball as the volume
of a hummingbird eyeball?
hummingbird: human:
The human eyeball is about 72.3 times as great
in volume as a hummingbird eyeball.

15. In the figure, the vertex of the pyramid is at the
center of the sphere. The height of the pyramid is
approximately the radius r of the sphere. Suppose the
entire sphere is filled with n pyramids that each have
base area B and height r.

16. 4πr2
≈ nB
If the pyramids fill the sphere, the total area of the
bases is approximately equal to the surface area of
the sphere S, so 4πr2
≈ S. As the number of pyramids
increases, the approximation gets closer to the actual
surface area.

18. Example 3A: Finding Surface Area of Spheres
Find the surface area of a sphere with diameter
76 cm. Give your answers in terms of π.
S = 4πr2
S = 4π(38)2
= 5776π cm2
Surface area of a sphere

19. Example 3B: Finding Surface Area of Spheres
Find the volume of a sphere with surface area
324π in2
. Give your answers in terms of π.
Substitute 324π for S.324π = 4πr2
r = 9 Solve for r.
Substitute 9 for r.
The volume of the sphere is 972π in2
.
S = 4πr2
Surface area of a sphere

20. Example 3C: Finding Surface Area of Spheres
Find the surface area of a
sphere with a great circle
that has an area of 49π mi2
.
Substitute 49π for A.49π = πr2
r = 7 Solve for r.
S = 4πr2
= 4π(7)2
= 196π mi2
Substitute 7 for r.
A = πr2
Area of a circle

21. Check It Out! Example 3
Find the surface area of the sphere.
Substitute 25 for r.
S = 2500π cm2
S = 4πr2
S = 4π(25)2
Surface area of a sphere

22. Example 4: Exploring Effects of Changing Dimensions
The radius of the sphere is
multiplied by . Describe the
effect on the volume.
original dimensions: radius multiplied by :
Notice that . If the radius is multiplied
by , the volume is multiplied by , or .

23. Check It Out! Example 4
The radius of the sphere is
divided by 3. Describe the
effect on the surface area.
original dimensions: dimensions divided by 3:
The surface area is divided by 9.
S = 4πr2
= 4π(3)2
= 36π m3
S = 4πr2
= 4π(1)2
= 4π m3

24. Example 5: Finding Surface Areas and Volumes of
Composite Figures
Find the surface area and
volume of the composite figure.
Give your answer in terms of π.
Step 1 Find the surface area of the
composite figure.
The surface area of the composite figure is the
sum of the curved surface area of the
hemisphere, the lateral area of the cylinder, and
the base area of the cylinder.

25. Example 5 Continued
The surface area of the composite figure is
L(cylinder) = 2πrh = 2π(6)(9) = 108π in2
B(cylinder) = πr2
= π(6)2
= 36π in2
72π + 108π + 36π = 216π in2
.
Find the surface area and
volume of the composite figure.
Give your answer in terms of π.

26. Step 2 Find the volume of
the composite figure.
Example 5 Continued
Find the surface area and
volume of the composite figure.
Give your answer in terms of π.
The volume of the composite figure is
the sum of the volume of the
hemisphere and the volume of the
cylinder.
The volume of the composite figure is 144π + 324π = 468π in3
.

27. Check It Out! Example 5
Find the surface area and
volume of the composite figure.
Step 1 Find the surface area of the
composite figure.
The surface area of the composite figure is the
sum of the curved surface area of the
hemisphere, the lateral area of the cylinder, and
the base area of the cylinder.

28. Check It Out! Example 5 Continued
The surface area of the composite figure is
Find the surface area and
volume of the composite figure.
L(cylinder) = 2πrh = 2π(3)(5) = 30π ft2
B(cylinder) = πr2
= π(3)2
= 9π ft2
18π + 30π + 9π = 57π ft2
.

29. Step 2 Find the volume of the
composite figure.
Find the surface area and
volume of the composite figure.
Check It Out! Example 5 Continued
The volume of the composite figure is the volume of
the cylinder minus the volume of the hemisphere.
V = 45π – 18π = 27π ft3

30. Lesson Quiz: Part I
Find each measurement. Give your answers in
terms of π.
1. the volume and surface area
of the sphere
2. the volume and surface area of a sphere with
great circle area 36π in2
3. the volume and surface area
of the hemisphere
V = 36π cm3
;
S = 36π cm2
V = 288π in3
;
S = 144π in2
V = 23,958π ft3
;
S = 3267π ft2

31. Lesson Quiz: Part II
4. A sphere has radius 4. If the radius is multiplied
by 5, describe what happens to the surface area.
5. Find the volume and surface area of the
composite figure. Give your answer in terms of π.
The surface area is multiplied by 25.
V = 522π ft3
; S = 267π ft2

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