Volume of Solid Figures

Section 3.8

Standard: MCC9-12.G.GMD.1-3

Essential Questions: How do I derive use the volume formulas, including Cavalieri’s Principle, to find the volume of cylinders, pyramids, cones, and spheres?

3.8A VOLUME OF A SPHERE

A sphere is formed by revolving a circle about its diameter.

In space, the set of all points that are a given distance from a given point, called the center.

Definition:

Formula: Volume of a Sphere

34

3V r

Spheres

1. Find the volume of the sphere, given that the radius is 8 inches.

34

3V r

34(8)

3V

2048

3V

V ≈ 2144.66 in.3

8

2. Find the volume of the sphere, given that the diameter is 10 inches.

34

3V r

34(5)

3V

500

3V

V ≈ 523.60 in.3

d = 2r10 = 2r 5 = r

10

3. Find the volume of the sphere, given that the circumference of the sphere is ft.8

C = 2r8 = 2r 4 = r

34

3V r

34(4)

3V 256

3V

V ≈ 268.08 ft3

Complete the table.

Sphere 1 Sphere 2 Ratio of each for Sphere 1: Sphere 2

Radius 2 3 2 : 3Diameter

Circumference

Area of Great Circle

Surface Area

Volume

4 6 4:6 or 2:34 6 4:6 or 2:3

4 9 4:9 or 4:9

16 36 16:36 or 4:9

32

3 36

32 108:

3 38: 27or

Complete the table.

Sphere 1 Sphere 2 Ratio of each for Sphere 1: Sphere 2

Radius a b a : bDiameter

Circumference

Area of Great Circle

Surface Area

Volume

2a 2b a : b2a 2b a : b

a2 b2 a2 : b2

4a2 4b2 a2 : b2

34

3a 34

3b a3 : b3

Scale Factor _____________

Area Ratios _______________

Volume Ratios _______________

a : b

a2 : b2

a3 : b3

4. Two spheres have diameters 24 and 36. a. What is the ratio of the areas?

b. What is the ratio of the volumes?

Ratio of volumes:

Radii:Ratio of radii:

Ratio of areas:

24 and 3624 : 36 or 2 : 3

22 : 32 or 4 : 9

23 : 33 or 8 : 27

5. A sphere has a radius of 6 meters. The radius of a second sphere is 3 meters. (a.) How does the surface area of the second sphere compare the to surface area of the first sphere? (b.) Volumes?

Ratio of radii: 3 : 6

(a). Ratio of areas: 12 : 22

or 1 : 2

or 1 : 4

The second sphere is ¼ the size of the first. (The area of the second sphere is 4 times smaller)

(b). Ratio of volumes: 13 : 23 or 1 : 8

The second sphere is 1/8 the size of the first. (The volume of second sphere is 8 times smaller)

6. The radius of a sphere is 2.4 cm. (a.) How will the surface area change if the radius is doubled? (b.) Volume?

Double the radius: 2(2.4) = 4.8

Ratio of radii: 2.4 : 4.8 or 1 : 2

Ratio of surface areas: 12 : 22 or 1 : 4

Ratio of volumes: 13 : 23 or 1 : 8

The surface area is 4 times larger if the radius is doubled.

The volume is 8 times larger if the radius is doubled.

7. Find the surface area of one hemisphere of a circle if the circumference of a great

circle of the sphere is 7 cm.

C = 2r7 = 2r7/2 = r

Surface area of entire sphere: S = 4r2

S = 4 (3.5)2

S = 49½S ≈ 76.93 cm2

Area of base of hemisphere: A = r2 A = (3.5)2

A = 12.25A ≈ 38.48 cm2

Surface area of hemisphere ≈ 76.93 cm2 + 38.48 cm2

≈ 115.41 cm2

3.8B VOLUME OF CYLINDERS, CONES,

PRISMS, & PYRAMIDS

2r h

Cylinders are right prisms with circular bases.Therefore, the formula for Volume can be used for cylinders.

Volume (V) = Bh =

Formula: V = 2r hCylinders

h

2πr

h

Example 8

For the cylinder shown, find the volume.

4 cm

3 cmV = πr2•hV = π(3)2•(4)

V = 36π

Cones

2r21 1

3 3Bh r h

Cones are right pyramids with a circular base.

Volume (V) =

The base area is the area of the circle:

Notice that the height (h) (altitude), the radius and the slant height create a right triangle.

Formulas: V = 21

3r h

l

r

h

6 cm

Example 9:

For the cone shown, find the volume.

V= 96π cubic cm.

2

2

1

31

6 83

V r h

V

8

Volume of a Right Prism (V )= Bh(h = height of prism, B = base area)

hP

hB

Triangular Right Prism

PRISMSPrism: A solid with Bases which are parallel and congruent polygons and Lateral faces which are parallelograms.

Example 10:

6

8

5

4

4

B = ½ (6)(4) = 12

V = 12 x 4 = 48 cubic units

h = 4

V= Bh

STICKY NOTE PROBLEMFind each volume to the nearest tenth. Use 3.14 for .

1. cylinder: radius = 6 m, height = 11 m

1,243.4 m3

1,114.6 cm3

612 ft3

Course 2

904.3 m3

2. rectangular prism: length = 10 cm, width = 8.64 cm, height = 12.9 cm

3. triangular prism: base area = 34 ft2, height = 18 ft

4. cylinder: diameter = 8 m, height = 18 m

Sticky Note Problem

definition:

the surface of a conic solid whose base is a polygon.

Lateral side

vertex

altitude

Slant height

Base

PYRAMIDS

Pyramid(B = base area)

The volume of a pyramid (V)= ⅓ Bh

Volume = ⅓ (100)(12) == 400 cubic units12

1010

13Example 11:

6

10

9

m

l

53

Sticky Note Problem

3.8C Cavalieri’s Principle

A cross section (of a geometric solid): the intersection of a plane and the solid.

A prism has the same cross section (parallel to the base) all along its length !

Shown here are the cross sections (in the same plane) of two prisms of equal height.  The cross section slices are indicated in red (grey in notes) and are parallel to the bases.

  If the areas of these two cross section slices are equal, the prisms will be equal in volume.

Cavalieri's Principle:  If, in two solids of equal height, the cross sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.

Seventeenth century mathematician, Bonaventura Cavalieri, generalized this concept for solids.

A generalized statement of this principle: Two prisms will have equal volumes if their bases have equal area and their altitudes (heights) are equal.

Use Cavalieri’s Principle to find the volume of the oblique prism.

Volume = (6X4)(3) == 72 mm3

V= Bh