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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

Slide 1 / 108

Volume

www.njctl.org

March 7, 2012

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Table of Contents

Volume of Pyramids

Volume of Prisms

Volume of Cylinders

Volume of Cones

Volume of a Sphere

Cavalieri's Principle

Corresponding Parts of Similar Solids

Coordinates in Space

Click on the topic to go to that section

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Volume of a Prism

Return to Table of Contents

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Definition of Volume -

The amount of cubic units that a solid can hold.

Where area used square units, volume will use cubic units.

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Base

height

Base

Lw

hV = Bh

Specific PrismsBox: V = LwhCube: V=s3

Finding the Volume of a Prism

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Does a prism need to be a right prism for the volume formula to work?

Think of a ream of paper

Stacked nicely it has 500 sheets.

If the stack is fanned, it still has 500 sheets.

So the volume doesn't change if the prism, stack of paper, is right or oblique.The formula V=BH works for all prisms

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Example: Find the volume of the box with length 2, width 6, and height 5.

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Example: The volume of a box is 48 ft3. If the height is 4ft and width is 6ft, what is the length.

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Example: Find the volume of the cube with edges of 7.

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Example: Find the volume of a cube is 64 m3, what is area of one face?

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Example: Find the volume of the prism with height 8 and hexagon base with apothem 4.

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1 What is the volume of a box with edges of 4, 5, and 7?

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2 What is the volume of a box with edges of 8, 6, and 10?

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3 What is the volume of a cube with edges of 5 units?

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4 If the volume of a box is 64 u3 and has height 8 and width 4, what is the length?

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5 If the volume of a box is 120 u3 and has height 6 and length 4, what is the width?

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6 If a cube has volume 27 u3, what is the cubes surface area?

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7 Find the volume of the prism.

15

1220

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8 Find the volume of the prism.

7

2

6

6

6

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Volume of a Cylinder

Return to Table of Contents

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Finding the Volume of a Cylinder

base

base

height

r

r

V = Bh

V = πr2h

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Example: Find the volume of the cylinder with radius 4 and height 11.

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Example: The surface area of a cylinder is 425 u2, what is the volume?

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9 Find the volume of the cylinder with radius 6 and height 8.

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10 Find the volume of the cylinder with circumference 18π units and height 6?

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11 Find the volume of the cylinder with a surface area 108 u2.

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12 The volume of a cylinder is 108 u3, what is the surface area?

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13 The height of a cylinder doubles, what happens to the volume?

A Doubles

B Quadruples

C Depends on the cylinder

D Cannot be determined

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14 The radius of a cylinder doubles, what happens to the volume?

A Doubles

B Quadruples

C Depends on the cylinder

D Cannot be determined

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15 A 3" hole is drilled through a solid cylinder with a diameter of 4" forming a tube. What is the volume of the tube?

24"

4"

3"

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Volume of a Pyramid

Return to Table of Contents

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Finding the Volume of a Pyramid

Square Base (B)

Slant Height (l )

Pyramid's Height (h)

V = 1/3 Bh

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Example: Find the volume of the pyramid.

54

6

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Example: Find the volume of the pyramid.

54

6

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Example: Find the volume of the pyramid.

88

5

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16 Find the volume of the pyramid.

76

5

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17 Find the volume of the pyramid.

66

8

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18 Find the volume of the pyramid.

12

12

10

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A truncated pyramid is a pyramid with its top cutoff parallel to its base.

Find the volume of the truncated pyramid shown.

22

66

9

3

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19 Find the volume of the pyramid.

22

88

12

3

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Volume of a Cone

Return to Table of Contents

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Finding the Volume of a Cone

r

height

Slant Height l

V = 1/3 Bh

V = 1/3 π r2 h

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Example: Find the volume of the cone.

9

r= 7

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Example: Find the volume of the cone.

12

r= 4

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Example: Find the volume of the cone, with lateral area 15π units2 and slant height 5.

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20 What is the volume of the cone?

8

d=10

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21 What is the volume of the cone?

10 40o

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22 What is the volume of the truncated cone?

r=8

r=4

6

6

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Volume of a Sphere

Return to Table of Contents

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Finding the Volume of a Sphere

rV = 4/3 π r3

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Example: Find the volume of the sphere.

9

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Example: Find the volume of the sphere.

Great Circle: A=25π u2

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Example: Find the volume of the sphere.

Surface Area: SA=36π u2

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23 Find the volume of the sphere.

4

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24 Find the volume of the sphere.

6

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25 Find the volume of the sphere.

Great Circle: A= 16π u2

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26 Find the volume of the sphere.

Surface Area: SA= 16π u2

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27 Find the volume of the sphere.

Cross Section: A= 16π u2

3

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Cavalieri's Principle

Return to Table of Contents

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Cavalieri's Principle

If two solids are the same height, and the area of their cross sections are equal, then the two solids will have the same volume

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14 14 14

Which solid has the greatest volume?

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A sphere is submerged in a cylinder. What is the volume of the cylinder not occupied by the sphere?

volume of cylinder - volume of sphere

The result shows that the left over volume is equal to 2 of what other solid?

According to Cavalieri, what can be said about the cross section?

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28 These 2 surfaces have the same volume, find x.

11 11

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29 These 2 surfaces have the same volume, find x.

12 12

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Corresponding Parts of Similar

Solids

Return to Table of Contents

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Corresponding parts of similar figures are similar.

The prisms shown are similar. Find x and y.

4 6

x9

y2

The ratio of similitude, k, is the common value that is multiplied to preimage to get to the image.

If the smaller prism is the preimage, what is k?

If the larger prism is the preimage, what is k?

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30 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of x.

8

8

16

h

2

x

y

3

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31 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of y.

8

8

16

h

2

x

y

3

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32 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of h.

8

8

16

h

2

x

y

3

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4 6

69

32

Consider the example of the prisms from earlier. The ratio of similitude from the smaller surface to the larger is 3/2.

Look at the area of their bases: how do they compare?

How do their volumes compare?

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Comparing Similar Figures

length in image

length in preimage= k

area in image

area in preimage = k2

volume in image

volume in preimage = k3

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Example:

r = 9r = 3

How many times bigger is the great circle on the right?

How many times bigger is the surface area on the right?

How many times bigger is the voklume on the right?

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33 The scale factor of 2 similar pyramids is 4. If the area of the base of the larger one is 64 u2, what is area of the smaller one?

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34 The scale factor of 2 similar right square pyramids is 3. If the area of the base of the larger one is 36 u2 and its height is 12, what is lateral area of the smaller one?

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35 An architect builds a scale model of a home using 2 in to 5 ft. scale. Given the view of the roof of the model, how much roofing material is needed for the house?

12in

6in8in

5in 4in

3in

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Coordinates in Space Graphing

Distance

Midpoint

Diagonal of a Box

Equation of a Sphere

Return to Table of Contents

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Graphing in space requires the x-, y-, and z-axes.z+ -

y+

-x+

-

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To graph an ordered triple, (x, y, z), draw a box with a vertice at the origin.

z+

-

y+

-x+

-

Graph (2, 4, 3)

(2,0,0)(2,4,0)

(0,4,0)

(0,4,3)(0,0,3)

(2,0,3)(2,4,3)

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To graph an ordered triple, (x, y, z), draw a box with a vertice at the origin.

z+

-

y+

-x+

-

Graph (-3, -1, -4)

(-3,0,0)

(0,-1,0)

(-3,-1,0)

(-3,0,-4)

(0,0,-4)(0,-1,-4)

(-3,-1,-4)

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z+ -

y+

-x+

-

Graph (2, 4, 9)

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z+ -

y+

-x+

-

Graph (-1, -4, 0)

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36 What is the ordered triple that was graphed?

A (2,3,4)

B (3,2,4)

C (4,3,2)

D (3,4,2)

z+ -

y+

-x+

-4

2

3

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37 What is the ordered triple that was graphed?

A (-2,3,4)

B (3,-2,4)

C (4,3,-2)

D (3,4,-2)

z+ -

y+

-x+

- -2

4

3

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Distance FormulaThe distance between two points in space:

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Example: Find the distance between (4,1,-5) and (-2, 8,-2)

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38 What is the distance between (4,-2,5) and (3,5,-6)?

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39 What is the distance between (-1,-2,-3) and (5,0,-4)?

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Midpoint SegmentThe midpoint of a segment is found by

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Example: Find the midpoint of (4,3,-8) and (-6,0,9)

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40 What is the midpoint of (4,8,10) and (6,4,-12)?

A (5,6,-1)

B (10,12,-2)

C (2,4,22)

D (1,2,11)

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41 What is the midpoint of (6,-1,5) and (6,5,-1)?

A (0,-6,4)

B (0,-3,2)

C (12,4,4)

D (6,2,2)

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Diagonal of a Box

h

wl

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Example: Find the diagonal of the box.

6

47

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42 Find the length of the diagonal.

8

5

1

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43 Find the length of the diagonal.

6

49

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Why is there no slope formula for 3 dimensional geometry?

If a line went through the point and had a slope of 4 what would that mean?

z+ -

y+

-x+

-

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SpheresRecall the definition of a circle:

Circle- the set of points in a plane a given distance from a given point.

The given distance was the radius and the given point was the center.

A sphere has a similar definition:

Sphere- the set of points a given distance from a given point.

The difference is a circle is a plane figure, a sphere is a space figure.

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The Equation of a SphereThe equation of a sphere is to that similar that of a circle.

Where (h,k,j) is the center and r is the radius.

{ (x,y,z) represent all of the ordered triples that lie on the sphere.}

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Given the equation (x-3)2 +(y+4)2+ z2 = 49

What is the center?

How long is the radius?

Is the point (4,2,2) inside, on, or outside the sphere?

(,3,-4,0)

7

(4-3)2 +(2+4)2+ (2)2 ? 49(1)2 +(6)2+ (2)2 ? 491+36+4 ? 4941<49 so inside

Name a point on the sphere.ex: (3,-4,7), (3,3,0), or (-4,-4,0)

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44 What is the center of

A

B

C

D

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45 What is the radius of

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46 What is the center of

A

B

C

D

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47 What is the radius of

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48 Is the point (0,0,0) inside, on, or outside the sphere with equation

A inside

B on

C outside

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49 Is the point (-4,6,3) inside, on, or outside the sphere with equation

A inside

B on

C outside

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50 What is the length of the radius of a sphere with equation

Hint: Complete the Square for x

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51 What is the length of the radius of a sphere with equation

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