CONFIDENTIAL 1

GeometryGeometry

Review Solid Review Solid GeometryGeometry

CONFIDENTIAL 2

Warm UpWarm Up

Determine whether the two polygons are similar. If so, give the similarity ratio.

1) 2)

8

8

2 2

12

4

12

4

11.942.5

40.8

7

24

25

CONFIDENTIAL 3

Solid GeometrySolid Geometry

Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the

intersection of two faces. A vertex is the point that is the intersection of three or more faces.

Face Edge

Vertex

CONFIDENTIAL 4

Three-Dimensional Figures Three-Dimensional Figures

TERM EXAMPLE

A Prism is formed by two parallel congruent polygonal faces called bases

connected by faces that are parallelograms.

Bases

A cylinder is formed by two parallel congruent circular bases and curved

surface that connects the bases. Bases

CONFIDENTIAL 5

TERM EXAMPLE

A pyramid is formed by a polygonal base and triangular faces that meet at

a common vertex.

Vertex

Base

A cone is formed by a circular base and a curved surface that connects

the base to a vertex.Base

Vertex

CONFIDENTIAL 6

A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of their bases.

TriangularPrism

RectangularPrism

PentagonalPrism

HexagonalPrism Next Page:

CONFIDENTIAL 7

Triangularpyramid

Rectangularpyramid

Pentagonalpyramid

Hexagonalpyramid

CONFIDENTIAL 8

Classifying Three-Dimensional Figures

Classify the figure. Name the vertices, edges, and bases.

A.

A

B C

D

E

Rectangular pyramid

Rectangular pyramid

Vertices: A,B,C,D,E

Edges: AB, BC, CD, AD, AE,BE, CE, DE

Base: rectangle ABCD

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CONFIDENTIAL 9

Identifying a Three-Dimensional Figure From a Net

Describe the three-dimensional figure that can be made from the given net.

A)

The net has two congruent triangular faces. The remaining faces are parallelograms, so the net forms a triangular prism.

CONFIDENTIAL 10

Describing Cross Sections of Three-Dimensional Figures

Describe the cross section.

A The cross section is a triangle.

CONFIDENTIAL 11

Food Application

A chef is slicing a cube-shaped watermelon for a buffet. How can the chef cut the watermelon to

make a slice of each shape?

A A square

Cut parallel to the bases.

CONFIDENTIAL 12

Now you try!

1) Classify each figure. Name the vertices, edges, and bases.

a)

B

Ab)

K

G

J

DC

EF

H

CONFIDENTIAL 13

Representations of Three-Dimensional Figures Representations of Three-Dimensional Figures

There are many ways to represent a three-dimensional object. An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side.

Top

Back

Bottom

Front

Left

Right

CONFIDENTIAL 14

Drawing Orthographic Views of an Drawing Orthographic Views of an ObjectObject

Draw all six orthographic views of the given object.Assume there are no hidden cubes.

Front:

Top:

Back:

Right:Left:

Bottom:

CONFIDENTIAL 15

Isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern.

CONFIDENTIAL 16

Drawing an isometric View of an Object

Draw an isometric view of the given object. assume there are no hidden cubes.

CONFIDENTIAL 17

In a perspective drawing, nonvertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing point are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing points. A two-point perspective drawing contains two vanishing points.

Vanishing point s

Vanishing point

one-point perspective

two-point perspective

CONFIDENTIAL 18

Drawing an Object in Perspective

3 A) Draw a cube in one-point perspective.

Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. This is the front of the cube.

From each corner of the square, lightly draw dashed segments to the vanishing point.

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CONFIDENTIAL 19

Lightly draw a smaller square with vertices on the dashed segments. This is the back of the cube.

Draw the edges of the cube, using dashed segments for hidden edges. Erase any segments that are not part of the cube.

Next page

CONFIDENTIAL 20

Relating Different Representations of an Object

4) Determine whether each drawing represents the given object. Assume there are no hidden cubes.

A B

Yes; the drawing is a one-point perspective view of the object.

No; the figure in the drawing is made up of four cubes, and the object is made up of only three cubes.

Next page

CONFIDENTIAL 21

Now you try!

1) Determine whether drawing represents the given object. Assume there are no hidden cubes.

Front

Top

BackRightLeft

Bottom

CONFIDENTIAL 22

Formulas in Three Dimensions

A polyhedron is formed by four or more polygons that intersect only at their edges. Prism and pyramids are

polyhedrons, but cylinders and cones are not.

polyhedrons Not polyhedron

CONFIDENTIAL 23

A polyhedron is a solid object whose surface is made up of a number of flat faces which themselves are bordered by straight lines. Each face is in fact a polygon, a closed shape in the flat 2-dimensional plane made up of points

joined by straight lines.

The familiar triangle and square are both polygons, but polygons can also have more irregular shapes like the one shown on the right.

A polygon is called regular if all of its sides are the same length, and all the angles between them are the same.

CONFIDENTIAL 24

A polyhedron is what you get when you move one dimension up. It is a closed, solid object whose surface is made up of a number of polygonal faces. We call the sides of these faces edges — two faces meet along each one of these edges. We call the corners of the faces vertices, so that any vertex lies on at least three

different faces. To illustrate this, here are two examples of well-known polyhedra.

The familiar cube on the left and the icosahedrons on the right. A polyhedron consists of polygonal faces, their sides are known as

edges, and the corners as vertices.

CONFIDENTIAL 25

A polyhedron consists of just one piece. It cannot, for example, be made up of two (or more) basically separate parts joined by

only an edge or a vertex. This means that neither of the following objects is a true polyhedron.

These objects are not polyhedra because they are made up of two separate parts meeting only in an edge (on the left) or a vertex (on the right).

CONFIDENTIAL 26

Euler’s Formula

For any polyhedron with V vertices, E edges, and F faces,

V - E + F = 2.

CONFIDENTIAL 27

Using Euler’s FormulaFind the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula.

A B

Using Euler’s Formula.Simplify

V= 4, E = 6, F = 44 - 6 + 4 = 2

2=2

V = 10, E = 15, F =710 - 15 + 7 = 2

2=2

CONFIDENTIAL 28

A diagonal of a three-dimensional figure connects

two vertices of two different faces. Diagonal d of a

rectangular prism is shown in the diagram. By the

PythagoreanTheoram, l + w = x and x + h = d.

Using substitution, l + w + h = d.

22 22

2

2

2 22

2

l

hd

wx

CONFIDENTIAL 29

Diagonal of a Right Rectangular Prism

The length of a diagonal d of a right rectangular prism with length l , width w, and height h is

d = 2 22

l + w + h .

CONFIDENTIAL 30

Space is the set of all points in three dimensions. Three coordinates are needed to locate a point in space. A three-dimensional coordinate system has 3 perpendicular axes: the x-axis, the y-axis, and the z-axis. An ordered triple (x , y ,z) is used to located a point. To located the point (3, 2, 4), start at (0, 0, 0). From there move 3 units forward, 2 units right, and then 4 units up

x

y

z8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

(3,2,4)

2

4

3

CONFIDENTIAL 31

Graphing Figures in Three Graphing Figures in Three DimensionsDimensions

Graph the figure.

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

(0, 0, 0)

(0, 0, 4)

(4, 0, 4)

(4, 0, 0)

x

z

y

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

(0, 4, 0)

(4, 4, 0)

A) A cubed with edge length 4 units and one vertex at (0, 0, 0)

The cube has 8 vertices:(0, 0, 0), (0, 4, 0),(0, 0, 4), (4, 0, 0)(4, 4, 0), (4, 0, 4), (0, 4, 4),(4, 4, 4)

Next page

CONFIDENTIAL 32

z

x

y

M(x2, y2, z2) (x1, y1, z1)

You can find the distance between the two points (x1, y1, z1)and (x2, y2, z2) by drawing a rectangular prism with the given points as endpoints of a diagonal. Then use the formula for the length of the diagonal. You can also use a formula related to the Distance formula. (see Lesson 1-6.) The formula for the midpoint between (x1, y1, z1) and (x2, y2, z1) is related to the Midpoint formula. (see Lesson 1-6.)

CONFIDENTIAL 33

Distance and Midpoint Formulas in three Dimensions

The distance between the points (x1, y1, z1) and (x2, y2, z2) is d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2 .

The midpoint of the segment with endpoints (x1, y1, z1) and(x2, y2, z2) is

M(x1 + x2)

2,

(y1+ y2)

2,

(z1 + z2)

2 .

CONFIDENTIAL 34

Finding Distances and Midpoints in Three Dimensions

Find the distance between the given points. Find the midpoint of the segment with the given endpoints.

Round to the nearest tenth, if necessary.

A) (0, 0, 0) and (3, 4, 12)

Distance: Midpoint:

d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2

= (3 - 0)2 + (4 - 0)2 + (12 - 0)2

= 9 + 16 + 144 = 169 = 13 units

M(x1 + x2)

2,

(y1+ y2)

2,

(z1 + z2)

2

M0 + 3

2,

0 + 4

2,

0 + 12

2

M(1.5, 2, 6)

CONFIDENTIAL 35

Recreation Application

Two divers swam from a boat to the locations shown in the diagram. How far apart are the divers?

9 ft

Depth: 8 ft

Depth: 12 ft

18 ft

15 ft

6 ft

The location of the boat can be represented by the ordered triple (0, 0, 0), and the location of the divers can be represented by the ordered triples (18, 9, -8) and (-15, -6, -12).

d = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2

= (-15 - 18)2 + (-6 - 9)2 + (-12+ 8)2

= 1330

= 36.5 units

Use the Distance Formula to find the distance between the divers.

CONFIDENTIAL 36

1) Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula.

Now you try!

a)b)

CONFIDENTIAL 37

Now you try!

2a. (0, 9, 5) and (6, 0, 12)

2b. (5, 8, 16) and (12, 16, 20)

Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round

to the nearest tenth, if necessary.

CONFIDENTIAL 38

Surface Area of Prisms and Cylinders

Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edges of a base. The lateral faces of a right prism are all rectangles. An oblique

prism has at least one nonrectangular lateral face.

Bases

Base edges

Bases

lateral faces

lateral edge

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CONFIDENTIAL 39

An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-

dimensional figure is the length of an altitude.

Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is

the sum of the areas of the lateral faces.

altitude

Next Page:

CONFIDENTIAL 40

Lateral Area and Surface Area of Right Prisms

The lateral area of a right prism with base perimeter P and height h is L = ph.

The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B.

The surface area of a cube with edge length s is S = 6s .

2

h

s

s

s

CONFIDENTIAL 41

The surface area of a right rectangular prism with length l, width w, and height h can be

written asS = 2lw + 2wh + 2lh.

CONFIDENTIAL 42

Finding Lateral Areas and surface Areas of Prisms

Find the lateral area and surface area of the right prism. Round to the nearest tenth, if necessary.

12 cm

6 cm8 cm

A) The rectangular prism L = ph = (28)12 = 336 cm P = 2(8) + 2(6) = 28 cm

S = Ph + 2 B = 336 + 2(6)(8) = 432 cm

2

2

CONFIDENTIAL 43

The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique

cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.

Bases

Bases

Axis

Axis

lateral surfaces

oblique cylinder right cylinder

CONFIDENTIAL 44

Lateral Area and Surface Area of Right Cylinders

The lateral area of a right cylinder with radius r and height h is L = 2 rh.

The surface area of a right cylinder with lateral area L and base area B is S = L + 2B, orS = 2 rh + 2 r .2

r

h

r

h

2 r

CONFIDENTIAL 45

Finding Lateral Areas and Surface Areas of Right Cylinders

A) Find the lateral area and surface area of the right cylinder. Give your answer in terms of .

2 m

5m

the radius is half the diameter, or 1 m.L = 2rh = 2(1)(5) = 10m2

S = L + 2r2 = 10 + 2(1)2 = 12m2

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CONFIDENTIAL 46

Finding Surface Areas of composite Three-Dimensional Figures

Find the surface area of the composite figure. Round to the nearest tenth.

4 ft

20 ft

16 ft

24 ft

The surface area of the right rectangular prism is

S = Ph + 2B = 80(20) + 2(24)(16) = 2368 ft2

CONFIDENTIAL 47

Exploring Effects of Changing dimensions

The length, width, and height of the right rectangular prism are doubled. Describe the effect on the surface

area.

3 in.

2 in.6 in.

Original dimensions:S = Ph + 2B = 16(3) + 2(6)(2) = 72 in 2

Length, width, and height doubled:S = Ph + 2B = 32(6) + 2(12)(4) = 288 in

2

Notice that 288 = 4(72). If the length, width, and height are doubled, the surface area is multiplied by 2, or 4.2

CONFIDENTIAL 48

Now you try!

1) Find the lateral area and surface area of a cube with edge length 8 cm.

CONFIDENTIAL 49

Now you try!

Use the information above to answer the following.

2) A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as

the pieces in the previous slide. Compare the surface areas. Which will melt faster?

3 cm

4 cm

2 cm

5 cm

5 cm

1 cm

CUBE 1CUBE 2

CONFIDENTIAL 50

Surface Area of Pyramids and Cones

The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.

regular pyramid nonregular pyramid

Slant height

Vertices

Lateral faces

Altitude

Bases

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CONFIDENTIAL 51

The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.

s

s s

sl

P = 4s

l

ssss

CONFIDENTIAL 52

Lateral and surface Area of a Lateral and surface Area of a regular pyramidregular pyramid

The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl.

The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B.

l

CONFIDENTIAL 53

Finding Lateral and surface Area of Pyramids

Find the lateral area and surface area of the pyramid.a.) a regular square pyramid with base edge length 5 in. and slant height 9in.

L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in.S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in

2

2 2 2

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CONFIDENTIAL 54

The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.

Slant height

right cone oblique cone

Vertices

Lateral surfaces

axis axis

Base

Next page:

CONFIDENTIAL 55

The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.

CONFIDENTIAL 56

Lateral and Surface Area of a right cone

Lateral and Surface Area of a right cone

r

l

rl

The lateral area of a right cone with radius r and slant height l is L = rl.

The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2

CONFIDENTIAL 57

Finding Lateral Area and Surface Area of right cones

Find the lateral area and surface area of the cone. Give your answers in terms of .

A) A right cone with radius 2 m and slant height 3 m.

L = rl Lateral area of a cone = (2) (3) = 6 m2 Substitute 2 for r and 3 for l.S = rl + r2 Surface area of a cone = 6 + (2)2 = 10 m2 Substitute 2 for r and 3 for l.

CONFIDENTIAL 58

Exploring Effects of Changing Dimensions

3 cm

5 cmThe radius and slant height of the right cone tripled. Describe the effect on the surface area.

Notice that 216 = 9(24). I f the radius and slant heightare tripled, the surface area is multiplied by 32, or 9.

S = rl + r2 = (9)(15) + (9)2 = 216 cm2

S = rl + r2 = (3)(5) + (3)2 = 24 cm2

Original dimensions: Radius and slant height tripled:

CONFIDENTIAL 59

Finding Surface Area of Composite Three-Dimensional Figures

Find the surface area of the composite figure.

28 cm

90

cm

45

cm

The height of the cone is 90 - 45 = 45 cm.By the Pythagorean Theorem,l = 282 + 452 = 53 cm. the lateral area of thecoan isL = rl = (28)(53) = 1484 cm2.

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CONFIDENTIAL 60

28 cm

90

cm

45

cmThe lateral area of the cylinder is L = 2rh = 2(28)(45) =2520 cm2.

The base area is B = r2 = (28)2 = 784 cm2

S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520 + 784 + 1484 = 4788 cm2

CONFIDENTIAL 61

Now you try!

1) Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10ft.

CONFIDENTIAL 62

Now you try!

16 ft

6 ft

2) Find the lateral area and surface area of the right cone.

CONFIDENTIAL 63

Volume of Prisms and Cylinders

A cube built out of 27 unit cubes has a volume of 27 cubic units.

The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill

the interior.

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CONFIDENTIAL 64

A right prism and an oblique prism with the same base and height have the same volume

Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-

sectional area at every level, they have the same volume.

CONFIDENTIAL 65

Volume of a Prism

h

B B

h

The volume of a prism with base area B and height h is V = Bh.

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CONFIDENTIAL 66

s

s

s

h

w

l

The volume of a right rectangular prism with length l, width w, and height h is V = lwh.

The volume of a cube with edge length s is V = s . 3

CONFIDENTIAL 67

Finding Volumes of Prisms

Find the volume of the prism. Round to the nearest tenth, if necessary.

A).

10 cm

12 cm

8 cm

volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh.

V = lwh = (10)(12)(8) = 980 cm

3

CONFIDENTIAL 68

Marine Biology Application

The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in

gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.

(Hint: 1 gallon = 0.134 ft )3

8 ft

120 ft

60 ft

Next Page:

CONFIDENTIAL 69

8 ft

120 ft

60 ft

Step:1Step:1 Find the volume of the aquarium in cubic feet.

V = lwh = (120)(60)(8) = 57,600 cm 3

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CONFIDENTIAL 70

8 ft

120 ft

60 ft

Step:2Step:2 Use the conversion factor 1 gallon

0.134 ft3 to estimate the

volume in gallons.

57,600 ft3 1 gallon

0.134 ft3 = 429,851 gallons

1 gallon

0.134 ft3 = 1

Next Page:

CONFIDENTIAL 71

Step:3Step:3

Use the conversion factor 8.33 pounds

1 gallon to estimate the weight of the

water.

429,851 gallons 8.33 pounds

1 gallon 3,580,659 pounds

8.33 pounds

1 gallons = 1

The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds

8 ft

120 ft

60 ft

CONFIDENTIAL 72

Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they

have the same volume.

CONFIDENTIAL 73

r

h

r

h

The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h.2

CONFIDENTIAL 74

12 cm

8 cm

Find the volume of the cylinder. Give your answers both in terms of and rounded to the nearest tenth.

A).

V = r2h Volume of a cylinder = 8 2 12 Substitute 8 for r and 12 for h. = 768 cm3 2412.7 cm3

Next Page:

Finding Volumes of Cylinders

CONFIDENTIAL 75

Exploring Effects of Changing Dimensions

The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume.

6 m

12 moriginal dimensions: radius and height multiplied by1

2.

V = r2h V = r2h = (6)2(12) = (3)2(6) = 432 m3 =54 m3

Notice that 54 = 1

8(432). if the radius and height are multiplied by

1

2, the volume is multiplied by

1

2 3

, or 1

8.

CONFIDENTIAL 76

Finding Volumes of Composite Three-Dimensional Figures

Find the volume of the composite figure. Round to the nearest tenth.

5 m

8 m

9 m

6 m

The base area of the prism is

B = 1

2(6)(8) =24 m2.

The volume of the prism is V = Bh = 24(9) = 216 m3.The cylinder's diameter equals the hypotenuse of the prism's base, 10 m. So the radius is 5 m.The volume of the cylinder is V = r2h = (5)2(5) = 125m3.The total volume of the figure is the sum of the volumes.V = 216 + 125 608.7 m3

CONFIDENTIAL 77

Now you try!

1) Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with

legs 7 yd and 5 yd long.

CONFIDENTIAL 78

Now you try!

2) The length, width, and height of the prism are doubled. Describe the effect on the volume.

1.5 ft

4 ft 3 ft

CONFIDENTIAL 79

Volume of Pyramids and Cones

The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown.

Next Page:

CONFIDENTIAL 80

The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube.

CONFIDENTIAL 81

Volume of a Pyramid

The volume of a pyramid with base area B and height h is V = 1/3 Bh.

h h

BB

CONFIDENTIAL 82

Finding Volumes of Pyramids

Next Page:

4 in.

4 in.

6 in.

Find the volume of each pyramid.

a)A rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft.

b) The square pyramid the base is a square with a side length of 4 in., and the height is 6 in.

V = 1

3Bh =

1

3(42)(6) = 32 in3

V = 1

3Bh =

1

3(7 9)(12) = 252 ft3

CONFIDENTIAL 83

Architecture Application

The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd, 1 story = 10 ft)

2

1 acre

10 storiesThe base is a squarewith an area of about4840 yd2. the base edgelength is 4840 = 70 yd.the height is about10(10) = 100 ft, orabout 33 yd.

Next Page:

First find the volume in cubic yards.

V = 1

3Bh Volume of a regular pyramid

= 1

3(702)(33) = 53,900 yd3 Substitute 702 for B and 33 for h.

CONFIDENTIAL 84

1 acre

10 stories

Then convert your answer to find the volume in cubic feet. thevolume of one cubic yard is (3 ft)(3 ft)(3 ft) = 27 ft3.

Use the conversion factor 27 yd3

1 yd3 to find the volume in cubic feet.

53,900 yd3 27 yd3

1 yd3 1,455,300 ft3

CONFIDENTIAL 85

Volume of a Cones

h

r

h

r

The volume of a cone with base area B, radius r,

and height h is V = 1

3Bh, or V =

1

3r2h.

CONFIDENTIAL 86

Finding Volumes of a Cones

Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth.

A) A cone with radius 5 cm and height 12 cm

Next Page:

V =1

3r2h Volume of a cone

=1

3(5)2 (12) Substitute 5 for r and 12 for h.

= 100 cm3 314.2 cm3 Simplify.

CONFIDENTIAL 87

B) A cone with a base circumference of 21 cm and a height 3 cm less than twice the radius

Step 1: Use the circumference to find the radius.

Step 2: Use the radius to find the height.

2(10.5) – 3 = 18 cm The height is 3 cm less than twice the radius.

Step 3: Use the radius and height to find the volume.

2r = 21 Substitute 21 for C. r = 10.5 cm Divide both sides by 2.

V = 1

3 r2h Volume of a cone

= 1

3(10.5)2 (18) Substitute 10.5 for r and 18 for h.

=661.5 cm3 2078.2 cm3 Simplify.Next Page:

CONFIDENTIAL 88

25 ft

7 ft

Step 1: Use the Pythagorean Theorem to find the height.

Step 2: Use the radius and height to find the volume.

V = 1

3 r2h Volume of a cone

= 1

3(7)2 (24) Substitute 7 for r and 24 for h.

=392 ft3 1231.5 ft3 Simplify.

72 + h2 = 252 Pythgorean Theorem h2 = 576 Subtract 72 from both sides. h = 24 Take the square root of both sides.

CONFIDENTIAL 89

Exploring Effects of Changing Dimensions

The length, width, and height of the rectangular pyramid are multiplied by ¼ . Describe the effect on the volume.

20 ft

24 ft20 ft

Next Page:

CONFIDENTIAL 90

20 ft

24 ft20 ft

Length, width, and height multiplied by ¼:

Original dimensions:

V = 1

3Bh

= 1

3(24 20)(20)

= 3200 ft3

V = 1

3Bh

= 1

3(6 5)(5)

= 50 ft3

Notice that 50 = 1

64(3200). I f the length, width, and height

are multiplied by 1

4, the volume is multiplied by

1

4 3

, or 1

64.

CONFIDENTIAL 91

Finding Volumes of Composite Three-Dimensional Figures

Find the volume of the composite figure. Round to the nearest tenth.

2 in

4 in

5 in

The volume of the cylinder is V = r2h = 2 2 (2) = 8 in3.The volume of the cone is

V= 1

3r2h =

1

3 2 2(3) = 4 in3.

The volume of the composite figure isthe sum of the volumes. V= 8 + 4 = 12 in3 37.7 in3

CONFIDENTIAL 92

Now you try!

1) Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of

the base.

CONFIDENTIAL 93

Now you try!

2) The radius and height of the cone are doubled. Describe the effect on the volume.

9 cm

18 cm

CONFIDENTIAL 94

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 95

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 96

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 97

Volume of a Sphere

r

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

3 r3.

CONFIDENTIAL 98

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 99

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 100

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 101

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 102

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 103

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 104

r

Surface Area of a Sphere

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

CONFIDENTIAL 105

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.

CONFIDENTIAL 106

Now you try!

50 cm

3) Find the surface area of the sphere.

CONFIDENTIAL 107

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 108

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 109

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 110

Now you try!

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

CONFIDENTIAL 111

Now you try!

3 ft

5 ft

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

CONFIDENTIAL 112

You did a You did a greatgreat job job today!today!