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 425 in 3 the area of circle T if the diameter is 26 ft ID: 709673
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Slide1
Volume and surface area of a sphereSlide2
Warm Up
Find each measurement.
1. the radius of circle M if the diameter is 25 cm2. 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 625 cm2
12.5 cm
85 in.
169 ft2
50
cmSlide3
Learn and apply the formula for the volume of a sphere.
Learn and apply the formula for the surface area of a sphere.
ObjectivesSlide4
sphere
center of a sphere
radius of a spherehemispheregreat circleVocabularySlide5
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 hemispheresSlide6
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.Slide7
V
(hemisphere) =
V(cylinder) – V(cone)The volume of a sphere with radius r is twice the volume of the hemisphere, or .Slide8Slide9
Example 1A: Finding Volumes of Spheres
Find the volume of the sphere. Give your answer in terms of
.
= 2304
in3
Simplify.
Volume of a sphere.Slide10
Example 1B: Finding Volumes of Spheres
Find the diameter of a sphere with volume 36,000
cm3.
Substitute 36,000
for V.
27,000 = r3r = 30
d
= 60 cm
d = 2r
Take the cube root of both sides.
Volume of a sphere.Slide11
Example 1C: Finding Volumes of Spheres
Find the volume of the hemisphere.
Volume of a hemisphere
Substitute 15 for r.
= 2250
m
3
Simplify.Slide12
Check It Out!
Example 1
Find the radius of a sphere with volume 2304 ft
3.Volume of a sphere
Substitute for V.
r = 12 ft
Simplify.Slide13
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.Slide14
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. Slide15
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.Slide16
4
r2 ≈ nBIf the pyramids fill the sphere, the total area of the bases is approximately equal to the surface area of the sphere S, so 4r2 ≈
S. As the number of pyramids increases, the approximation gets closer to the actual surface area.Slide17Slide18
Example 3A: Finding Surface Area of Spheres
Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of
.
S = 4r2
S = 4(38)2 = 5776
cm2
Surface area of a sphereSlide19
Example 3B: Finding Surface Area of Spheres
Find the volume of a sphere with surface area 324
in2. Give your answers in terms of .
Substitute 324
for S.
324 = 4r2
r
= 9
Solve for r.
Substitute 9 for r.
The volume of the sphere is 972
in
2
.
S
= 4
r
2
Surface area of a sphereSlide20
Example 3C: Finding Surface Area of Spheres
Find the surface area of a sphere with a great circle that has an area of 49
mi2.
Substitute 49
for A.49
=
r
2
r
= 7
Solve for r.
S
= 4
r
2
= 4
(7)
2
= 196
mi
2
Substitute 7 for r.
A
=
r
2
Area of a circleSlide21
Check It Out!
Example 3
Find the surface area of the sphere.
Substitute 25 for r.
S = 2500
cm2S = 4r2
S
= 4
(25)
2
Surface area of a sphereSlide22
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 .Slide23
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
= 4
r
2
= 4
(3)
2
=
36
m
3
S
= 4
r
2
= 4
(1)
2
=
4
m
3Slide24
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
.
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. Slide25
Example 5 Continued
The surface area of the composite figure is
L
(cylinder) = 2rh = 2(6)(9) = 108
in2B(cylinder) = r
2 = (6)2 = 36 in2
72
+ 108
+
36
= 216
in
2
.
Find the surface area and volume of the composite figure. Give your answer in terms of
.Slide26
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 .
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 144 + 324 = 468
in
3
.
Slide27
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. Slide28
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) = 2rh = 2(3)(5) = 30 ft2
B
(cylinder) =
r
2
=
(3)
2
= 9
ft
2
18
+ 30
+
9
= 57
ft
2
.Slide29
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 = 45 – 18 = 27
ft
3Slide30
Lesson Quiz: Part I
Find each measurement. Give your answers in terms of
.1. the volume and surface area of the sphere 2. the volume and surface area of a sphere with great circle area 36 in23. the volume and surface area of the hemisphere
V
= 36 cm3; S = 36 cm2
V
=
288
in
3
;
S
=
144
in
2
V
=
23,958
ft
3
;
S
=
3267
ft
2Slide31
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 .
The surface area is multiplied by 25.
V = 522 ft3; S = 267 ft2