CONFIDENTIAL 1

GeometryGeometry

Spheres Spheres

CONFIDENTIAL 2

Warm UpWarm Up

Describe the effect on the volume that results from the given change.

1)The side length of a cube are multiplied by ¾.

2)The height and the base area of a prism are multiplied by 5.

1) the volume is decreased by 27/64 times.2) the volume is increased by 25 times.

CONFIDENTIAL 3

Spheres

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 to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres.

Center

HemisphereGreat circle

Radius

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CONFIDENTIAL 4

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.

h

r

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h

r

CONFIDENTIAL 5

h

r

V(hemisphere) = V(cylinder) - V(cone)

= r2h - 1

3 r2h

= 2

3 r2h

= 2

3 r2(r) The height of the hemisphere is equal

to the radius.

= 2

3 r3

The volume of a sphere with radius r is twice the volume of

the hemisphere, or V = 4

3 r3.

h

r

CONFIDENTIAL 6

Volume of a Sphere

r

The volume of a sphere with radius r is V = 4

3 r3.

CONFIDENTIAL 7

Finding Volumes of Spheres

9 cm

Find each measurement. Give your answer in terms of .

A) The volume of the sphere

V = 4

3 r3

V = 4

3 (9)2 Substitute 9 for r.

= 972 cm2 Simplify.

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CONFIDENTIAL 8

9 cm

972 = 4

3 r3 Substitute 972 for V.

729 = r3 Divide both sides by 4

3.

r = 9 Take the cube root of both sides.

d = 18 in. d = 2r

B) The diameter of a sphere with volume 972 in 3

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CONFIDENTIAL 9

4 m

C) The volume of the hemisphere

V = 2

3 r3 Volume of a hemisphere

= 2

3 4 3 =

128

3 m3 Substitute 4 for r.

CONFIDENTIAL 10

Now you try!

1) Find the radius of a sphere with volume 2304 ft .3

1) 11.7 ft

CONFIDENTIAL 11

Biology Application

Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximate a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximate a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball?

human eyeball: giant squid eyeball:

V = 4

3r3 V =

4

3r3

=4

3(1.25)3 8.18 cm3 =

4

3(12.5)3 8181.23 cm3

A giant squid eyeball is about 1000 times as great in volume as a human eyeball.

CONFIDENTIAL 12

Now you try!

2) A hummingbird eyeball has a diameter of approximately 0.625 cm. How many times as great is the

volume of a human eyeball as the volume of a hummingbird eyeball. A human eyeball is approximate a

sphere with a diameter of 2.5 cm. ?

2) the volume of human eye ball is 64 times the volume of humming bird.

CONFIDENTIAL 13

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

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CONFIDENTIAL 14

V(sphere) 1

3Br +

1

3Br + ......+

1

3Br The sphere's volume is close to the

sum of the volumes of the pyramids.

4

3r3 n

1

3Br

4r2 nB Divide both sides by 1

3r.

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CONFIDENTIAL 15

If the pyramids fill the sphere, the total area of the bases is approximate equal to the surface area of the sphere S, so4 r = S. As the number of pyramids increases, the approximation gets closer to the actual surface area.

2

CONFIDENTIAL 16

r

Surface Area of a Sphere

The surface area of a sphere with radius r is S = 4 r .2

CONFIDENTIAL 17

Finding Surface Area of Spheres

Find each measurement. Give your answers in terms of .

A) the surface area of a sphere with diameter 10 ft.

S = 4r2

S = 4(5)2 = 200 ft2 Substitute 5 for r.

B) the volume of a sphere with surface area 144 m2

S = 4r2

144 = 4r2 Substitute 144 for S. 6 = r Solve for r.

V = 4

3r3

= 4

3 6 3 = 288 m3 Substitute 6 for r.

The volume of the sphere is 288 m3.Next Page:

CONFIDENTIAL 18

A = 4 in2

C) the surface area of a sphere with a great circle that has an area of 4 in2

r2 = 4 Substitute 4 for A in the formula for the area of a circle. r = 2 Solve for r. S = 4r2

= 4(2)2 = 16 in2 Substitute 2 for r in the surface area formula.

CONFIDENTIAL 19

Now you try!

50 cm

3) Find the surface area of the sphere.

3) 2500∏ cm2

CONFIDENTIAL 20

Exploring Effects of Changing Dimensions

The radius of the sphere is tripled. Describe the effect on the volume.

3 m

Original dimensions: radius tripled:

V = 4

3r3 V =

4

3r3

= 4

3 3 3 =

4

3 9 3

= 36 m3 = 972 m3

Notice that 972 = 27(36). I f the radius is tripled,the volume is multiplied by 27.

CONFIDENTIAL 21

Now you try!

4) The radius of the sphere above is divided by 3. Describe the effect on the surface area.

4) the volume decrease by 9 times.

CONFIDENTIAL 22

Finding Surface Areas and Volumes of Composite Figures

Find the surface area and volume of the composite figure. Give your answers in terms of .

7 cm

25 cm

Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone.

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S(hemisphere) = 1

2(4r2) = 2 7 2 = 98 cm2

L(cone) = rl = (7)(25) = 175 cm2

The surface area of the composite figure is 98 + 175 = 273 cm2.

CONFIDENTIAL 23

7 cm

25 cm

Step 2 Find the volume of the composite figure. First find the height of the cone.

h = 252 - 72 Pythagorean Theorem = 576 = 24 cm Simplify.

The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cone.

V(hemiphere) = 1

2

4

3r3 =

2

3 7 3 =

686

3 cm3

V(cone) = 1

3r2h =

1

3 7 2 24 = 392 cm3

The volume of the composite figureis 686

3 + 392 =

1862

3 cm3.

CONFIDENTIAL 24

Now you try!

3 ft

5 ft

5) Find the surface area and volume of the composite figure.

5) 57∏ ft2

CONFIDENTIAL 25

Now some problems for you to practice !

CONFIDENTIAL 26

Assessment

1)Find each measurement. Give your answers in terms of .

1 m11 in.

A) The volume of the hemisphere

B) The volume of the sphere

1a) 887.33 ∏ in3 1b) 1.33 ∏ m3.

CONFIDENTIAL 27

10 cm

5 cm

2)Approximately how many times as great is the volume of the grapefruit as the volume of the lime?

2) 8 times

CONFIDENTIAL 28

16 yd

3)Find each measurement. Give your answers in terms of .

A = 49 cm2

A) The surface area of the sphere

B) The surface area of the sphere

3a) 256∏ yd2 3b) 196∏ cm2.

CONFIDENTIAL 29

15 in.16 cm

4) Describe the effect of each change on the given measurement of the figure.

A) Surface area The dimensions are doubled.

B) Volume The dimensions are multiplied by ¼.

4a) Increases by 4 times4b) Decreases by 64 times.

CONFIDENTIAL 30

5) Find the surface area and volume of the composite figure.

5 ft 2 ft

5a) SA = 36∏ ft3 V = 30.67∏ ft3

CONFIDENTIAL 31

Let’s review

CONFIDENTIAL 32

Spheres

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 to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres.

Center

HemisphereGreat circle

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CONFIDENTIAL 33

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.

h

r

Next Page:

h

r

CONFIDENTIAL 34

h

r

V(hemisphere) = V(cylinder) - V(cone)

= r2h - 1

3 r2h

= 2

3 r2h

= 2

3 r2(r) The height of the hemisphere is equal

to the radius.

= 2

3 r3

The volume of a sphere with radius r is twice the volume of

the hemisphere, or V = 4

3 r3.

h

r

CONFIDENTIAL 35

Volume of a Sphere

r

The volume of a sphere with radius r is V = 4

3 r3.

CONFIDENTIAL 36

Finding Volumes of Spheres

9 cm

Find each measurement. Give your answer in terms of .

A) The volume of the sphere

V = 4

3 r3

V = 4

3 (9)2 Substitute 9 for r.

= 972 cm2 Simplify.

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CONFIDENTIAL 37

9 cm

972 = 4

3 r3 Substitute 972 for V.

729 = r3 Divide both sides by 4

3.

r = 9 Take the cube root of both sides.

d = 18 in. d = 2r

B) The diameter of a sphere with volume 972 in 3

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CONFIDENTIAL 38

4 m

C) The volume of the hemisphere

V = 2

3 r3 Volume of a hemisphere

= 2

3 4 3 =

128

3 m3 Substitute 4 for r.

CONFIDENTIAL 39

Biology Application

Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximate a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximate a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball?

human eyeball: giant squid eyeball:

V = 4

3r3 V =

4

3r3

=4

3(1.25)3 8.18 cm3 =

4

3(12.5)3 8181.23 cm3

A giant squid eyeball is about 1000 times as great in volume as a human eyeball.

CONFIDENTIAL 40

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

Next Page:

CONFIDENTIAL 41

V(sphere) 1

3Br +

1

3Br + ......+

1

3Br The sphere's volume is close to the

sum of the volumes of the pyramids.

4

3r3 n

1

3Br

4r2 nB Divide both sides by 1

3r.

Next Page:

CONFIDENTIAL 42

If the pyramids fill the sphere, the total area of the bases is approximate equal to the surface area of the sphere S, so4 r = S. As the number of pyramids increases, the approximation gets closer to the actual surface area.

2

CONFIDENTIAL 43

r

Surface Area of a Sphere

The surface area of a sphere with radius r is S = 4 r .2

CONFIDENTIAL 44

Finding Surface Area of Spheres

Find each measurement. Give your answers in terms of .

A) the surface area of a sphere with diameter 10 ft.

S = 4r2

S = 4(5)2 = 200 ft2 Substitute 5 for r.

B) the volume of a sphere with surface area 144 m2

S = 4r2

144 = 4r2 Substitute 144 for S. 6 = r Solve for r.

V = 4

3r3

= 4

3 6 3 = 288 m3 Substitute 6 for r.

The volume of the sphere is 288 m3.Next Page:

CONFIDENTIAL 45

A = 4 in2

C) the surface area of a sphere with a great circle that has an area of 4 in2

r2 = 4 Substitute 4 for A in the formula for the area of a circle. r = 2 Solve for r. S = 4r2

= 4(2)2 = 16 in2 Substitute 2 for r in the surface area formula.

CONFIDENTIAL 46

Exploring Effects of Changing Dimensions

The radius of the sphere is tripled. Describe the effect on the volume.

3 m

Original dimensions: radius tripled:

V = 4

3r3 V =

4

3r3

= 4

3 3 3 =

4

3 9 3

= 36 m3 = 972 m3

Notice that 972 = 27(36). I f the radius is tripled,the volume is multiplied by 27.

CONFIDENTIAL 47

Finding Surface Areas and Volumes of Composite Figures

Find the surface area and volume of the composite figure. Give your answers in terms of .

7 cm

25 cm

Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone.

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S(hemisphere) = 1

2(4r2) = 2 7 2 = 98 cm2

L(cone) = rl = (7)(25) = 175 cm2

The surface area of the composite figure is 98 + 175 = 273 cm2.

CONFIDENTIAL 48

7 cm

25 cm

Step 2 Find the volume of the composite figure. First find the height of the cone.

h = 252 - 72 Pythagorean Theorem = 576 = 24 cm Simplify.

The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cone.

V(hemiphere) = 1

2

4

3r3 =

2

3 7 3 =

686

3 cm3

V(cone) = 1

3r2h =

1

3 7 2 24 = 392 cm3

The volume of the composite figureis 686

3 + 392 =

1862

3 cm3.

CONFIDENTIAL 49

You did aYou did a great great job job today!today!