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Geometry

Spheres

CONFIDENTIAL 1

Warm 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 2

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

Great circle Hemisphere Radius CONFIDENTIAL Center Next Page: 3

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 h r CONFIDENTIAL Next Page: 4

h r h V(hemisphere) = V(cylinder) - V(cone) =  r 2 h - 1 3  r 2 h = 2 3  r 2 h = 2 3  r 2 (r) The height of the hemisphere is equal to the radius.

= 2 3  r 3 The volume of a sphere with radius r is twice the volume of the hemisphere, or V = 4 3  r 3 .

r CONFIDENTIAL 5

Volume of a Sphere

The volume of a sphere with radius r is V = 4 3  r 3 .

r CONFIDENTIAL 6

Finding Volumes of Spheres

Find each measurement. Give your answer in terms of .

A) The volume of the sphere V = 4 3  r 3 V = 4 3  (9) 2 Substitute 9 for r.

= 972  cm 2 Simplify.

9 cm CONFIDENTIAL Next Page: 7

B) The diameter of a sphere with volume 972 in

9 cm 972  = 4 3  r 3 Substitute 972  for V.

729 = r 3 Divide both sides by 4 3  .

r = 9 Take the cube root of both sides.

d = 18 in. d = 2r Next Page: CONFIDENTIAL 8

C) The volume of the hemisphere 4 m V = 2 3  r 3 Volume of a hemisphere = 2 3  4 3 = 128 3  m 3 Substitute 4 for r. CONFIDENTIAL 9

Now you try!

1) 11.7 ft CONFIDENTIAL 10

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 3  r 3 V = 4 3 = 4 3  (1.25) 3  8.18 cm 3 = 4 3   r 3 (12.5) 3  8181.23 cm 3 A giant squid eyeball is about 1000 times as great in volume as a human eyeball.

CONFIDENTIAL 11

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 12

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 .

CONFIDENTIAL Next Page: 13

V(sphere)  1 3 Br + 1 3 Br + ......+ 1 3 Br The sphere's volume is close to the sum of the volumes of the pyramids.

4 3  r 3  n   4  r 2  nB Divide both sides by 1 3  r.

Next Page: 14 CONFIDENTIAL

If the pyramids fill the sphere, the total area of the bases is approximate equal to the surface area of the sphere S, so 4 r = S. As the number of pyramids increases, the approximation gets closer to the actual surface area. CONFIDENTIAL 15

Surface Area of a Sphere

r CONFIDENTIAL 16

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

B) the volume of a sphere with surface area 144  m 2 S = 4  r 2 144  = 4  r 2 Substitute 144  for S.

6 = r Solve for r.

V = 4 3  r 3 = 4 3  6 3 = 288  m 3 Substitute 6 for r.

The volume of the sphere is 288  m 3 .

CONFIDENTIAL Next Page: 17

C) the surface area of a sphere with a great circle that has an area of 4  in 2 A = 4  in 2  r 2 = 4 S = 4 = 4    r = 2 r 2 (2) 2 = 16  Substitute 4 Solve for r.

in 2  for A in the formula for the area of a circle.

Substitute 2 for r in the surface area formula.

CONFIDENTIAL 18

Now you try!

3) Find the surface area of the sphere.

50 cm 3) 2500∏ cm 2 CONFIDENTIAL 19

Exploring Effects of Changing Dimensions

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

Original dimensions: radius tripled: V = 4 3  r 3 V = 4 3  r 3 = 4 3  3 3 = 4 3  9 3 = 36  m 3 = 972  m 3 Notice that 972  = 27(36  ). If the radius is tripled, the volume is multiplied by 27.

3 m CONFIDENTIAL 20

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 21

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 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.

25 cm S(hemisphere) = 1 2 (4  r 2 ) = 2  7 2 = 98  cm 2 L(cone) =  r

=  (7)(25) = 175  cm 2 The surface area of the composite figure is 98  + 175  = 273  cm 2 .

CONFIDENTIAL Next Page: 22

Step 2 Find the volume of the composite figure.

First find the height of the cone.

h = 25 2 - 7 2 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) = V(cone) = 1 3 1 2    r 2 h = 1 3  7 3 2 =  24 2 3  7 3 = 686 3  = 392  cm 3  cm 3 The volume of the composite figureis 686  3 + 392  = 1862 3  cm 3 .

25 cm 7 cm CONFIDENTIAL 23

Now you try!

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

3 ft 5 ft 5) 57∏ ft 2 CONFIDENTIAL 24

Now some problems for you to practice !

CONFIDENTIAL 25

Assessment

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

A) The volume of the hemisphere B) The volume of the sphere 11 in.

1 m 1a) 887.33 ∏ in 3 1b) 1.33 ∏ m 3 .

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

5 cm 2) 8 times CONFIDENTIAL 27

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

A) The surface area of the sphere B) The surface area of the sphere 16 yd A = 49  cm 2 3a) 256∏ yd 2 3b) 196∏ cm 2 .

CONFIDENTIAL 28

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. 15 in.

16 cm 4a) Increases by 4 times 4b) Decreases by 64 times.

CONFIDENTIAL 29

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

5 ft 2 ft 5a) SA = 36∏ ft 3 V = 30.67∏ ft 3 CONFIDENTIAL 30

Let’s review

CONFIDENTIAL 31

Spheres

Great circle Hemisphere CONFIDENTIAL Center Next Page: 32

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 h r CONFIDENTIAL Next Page: 33

h r h V(hemisphere) = V(cylinder) - V(cone) =  r 2 h - 1 3  r 2 h = 2 3  r 2 h = 2 3  r 2 (r) The height of the hemisphere is equal to the radius.

= 2 3  r 3 The volume of a sphere with radius r is twice the volume of the hemisphere, or V = 4 3  r 3 .

r CONFIDENTIAL 34

Volume of a Sphere

The volume of a sphere with radius r is V = 4 3  r 3 .

r CONFIDENTIAL 35

Finding Volumes of Spheres

Find each measurement. Give your answer in terms of .

A) The volume of the sphere V = 4 3  r 3 V = 4 3  (9) 2 Substitute 9 for r.

= 972  cm 2 Simplify.

9 cm CONFIDENTIAL Next Page: 36

B) The diameter of a sphere with volume 972 in

9 cm 972  = 4 3  r 3 Substitute 972  for V.

729 = r 3 Divide both sides by 4 3  .

r = 9 Take the cube root of both sides.

d = 18 in. d = 2r Next Page: CONFIDENTIAL 37

C) The volume of the hemisphere 4 m V = 2 3  r 3 Volume of a hemisphere = 2 3  4 3 = 128 3  m 3 Substitute 4 for r. CONFIDENTIAL 38

Biology Application

CONFIDENTIAL 39

CONFIDENTIAL Next Page: 40

V(sphere)  1 3 Br + 1 3 Br + ......+ 1 3 Br The sphere's volume is close to the sum of the volumes of the pyramids.

4 3  r 3  n   4  r 2  nB Divide both sides by 1 3  r.

Next Page: 41 CONFIDENTIAL

Surface Area of a Sphere

r CONFIDENTIAL 43

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

B) the volume of a sphere with surface area 144  m 2 S = 4  r 2 144  = 4  r 2 Substitute 144  for S.

6 = r Solve for r.

V = 4 3  r 3 = 4 3  6 3 = 288  m 3 Substitute 6 for r.

The volume of the sphere is 288  m 3 .

CONFIDENTIAL Next Page: 44

C) the surface area of a sphere with a great circle that has an area of 4  in 2 A = 4  in 2  r 2 = 4 S = 4 = 4    r = 2 r 2 (2) 2 = 16  Substitute 4 Solve for r.

in 2  for A in the formula for the area of a circle.

Substitute 2 for r in the surface area formula.

CONFIDENTIAL 45

Exploring Effects of Changing Dimensions

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

3 m CONFIDENTIAL 46

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 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.

25 cm S(hemisphere) = 1 2 (4  r 2 ) = 2  7 2 = 98  cm 2 L(cone) =  r

=  (7)(25) = 175  cm 2 The surface area of the composite figure is 98  + 175  = 273  cm 2 .

CONFIDENTIAL Next Page: 47

Step 2 Find the volume of the composite figure.

First find the height of the cone.

h = 25 2 - 7 2 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.

25 cm 7 cm CONFIDENTIAL 48

You did a great job today!

CONFIDENTIAL 49