Warm-Up

Draw and name the 3-D solid of revolution formed by revolving the given 2-D shape around the x-axis.

Warm-Up

Draw and name the 3-D solid of revolution formed by revolving the given 2-D shape around the x-axis.

Sphere Hemisphere Torus

Volume and Surface Area of Spheres and Similar Solids

Objectives:

1. To derive and use the formulas for the volume and surface area of a sphere

2. To find the surface area and volume of similar solids

Sphere

A sphere is the set of all points in space at a fixed distance from a given point.

• Radius = fixed distance

• Center = given point

Exercise 1

What is the result of cutting a sphere with a plane that intersects the center of the sphere?

What 2-D shape is projected onto the plane?

Hemisphere

A hemisphere is half a sphere. The circle on the base of a hemisphere is a great circle.

Investigation 1

In this Investigation, we will discover the formula for the volume of a sphere. To do this we need to relate the sphere to a very particular cylinder.

Sphere Cylinder

Radius = r Radius = r

Height = 2r

Investigation 1

In this Investigation, we will discover the formula for the volume of a sphere. To do this we need to relate the sphere to a very particular cylinder.

Notice that this is the largest possible sphere that could fill the cylinder. This sphere is inscribed within the cylinder.

Investigation 1

Step 1: Rather than use the sphere, we’ll use the hemisphere with the same radius, since it will be easier to fill. So…fill the hemisphere.

Step 2: Pour the contents of the hemisphere into the cylinder. How full is it?

Investigation 1

Step 3: Repeat steps 1 and 2. How full is the cylinder?

Step 4: Repeat step 3. How full is the cylinder? What does this tell you about the volume of the sphere?

Archimedes Tomb

Archimedes was the first to discover that the volume of a sphere is 2/3 the volume of the cylinder that circumscribes it. He considered this to be his greatest mathematical achievement.

Exercise 2

Derive a formula for the volume of a sphere.

Sphere Cylinder

2

3V V

22

3r h

222

3r r

34

3r

h = 2r

Exercise 3

Derive a formula for the volume of a hemisphere.

Exercise 4

What is the extended ratio of the volume of the cone to the sphere to the cylinder?

Volume of Spheres and Hemispheres

Volume of a Sphere

• r = radius of the sphere

Volume of a Hemisphere

• r = radius of the hemisphere

343V r 32

3V r

Find the volume of each solid using the given measure.

1. d = 18.5 inches 2. C = 24,900 miles

Exercise 5

Find the volume of each solid using the given measures.

1. V = 2. V =

Exercise 6

Investigation 2

Now we’ll find a formula for the surface area of a sphere. To do this, perhaps we should use a net…

Or perhaps we’ll look at it another way.

Investigation 2

Think of a sphere as being constructed by a whole bunch of pyramids—I mean bunch of them. The height of each pyramid would be the radius of the sphere.n = a whole bunch

h = radius of the sphere B

Investigation 2

Let’s also say that each of these pyramids is congruent and has a base area of B.

Thus, the surface area of the sphere is:

1 2 3 nS B B B B (Not a very useful formula)

B

Investigation 2

Furthermore, the volume of the sphere should be the sum of the volumes of the pyramids.

1 1 1 11 2 33 3 3 3 nV B h B h B h B h

11 2 33 nV h B B B B

11 2 33 nV r B B B B

13V r S B

Exercise 7

Use the two formulas below to derive a formula for the surface area of a sphere.

13V r S

343V r

B

Exercise 8

Explain how the unwrapped baseball illustrates the formula for the surface area of a sphere.

Exercise 9

Derive a formula for the total surface area of a hemisphere.

SA of Spheres and Hemispheres

Surface Area of a Sphere

• r = radius of the sphere

Surface Area of a Hemisphere

• r = radius of the hemisphere

24S r 23S r

Find the surface area of each solid using the given measure.

1. d = 18.5 inches 2. C = 24,900 miles

Exercise 10

Similar Solids

Any two solids are similar solids if they are of the same type such that any corresponding linear measures (height, radius, etc.) have equal ratios.– Ratio = scale factor

Exercise 11

Explain why any two cubes are similar.

2"

4"

Exercise 12

Find the volume of a cube with a side length of 2 inches.

Now find the volume of a cube with a side length of 4 inches.

How do the volumes compare?

2"

4"

Exercise 12

Find the volume of a cube with a side length of 2 inches.

Now find the volume of a cube with a side length of 4 inches.

How do the volumes compare?

2"

4"

2" 2"

2" 2"

2" 2"

2" 2"

Volumes of Similar Figures

If two solids have a scale factor of a:b, then the corresponding volumes have a ratio of a3:b3.

Similarity Relationships

Perimeter Linear Units a:b

Area Square Units a2:b2

Volume Cubic Units a3:b3

For two shapes with a scale factor of a:b, each of the following relationships will be true.

Exercise 13

A breakfast-cereal manufacturer is using a scale factor of 5/2 to increase the size of one of its cereal boxes. If the volume of the original cereal box was 240 in.3, what is the volume of the enlarged box?

Exercise 14

Pyramids P and Q are similar. Find the scale factor of pyramid P to pyramid Q.

V = 1000 in3 V = 216 in3