CONFIDENTIAL 1

GeometryGeometry

Formulas in Three Formulas in Three DimensionsDimensions

1) cylinder 2) no edges3) O C

CONFIDENTIAL 2

Warm UpWarm Up

Use the diagram for Exercises 1-3.

C

D

1) Classify the figure.2) Name the edges.3) Name the base.

CONFIDENTIAL 3

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 4

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 5

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 6

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 7

In the lab before this lesson, you made a conjecture about the relationship between the

vertices, edges, and faces of a polyhedron. One way to state this relationship is given below.

CONFIDENTIAL 8

Euler’s Formula

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

V - E + F = 2.

CONFIDENTIAL 9

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 10

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

Now you try!

a) V= 6, E = 12, F = 86 - 12 + 8 = 22=2

a)b)

b) V= 7, E = 12, F = 77 - 12 + 7 = 22=2

CONFIDENTIAL 11

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 12

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 13

Using the Pythagorean Theorem in three Dimensions

Find the unknown dimension in each figure.

A)The length of the diagonal of a 3 in. by 4 in. by 5 in. rectangular prism

2 22d = 3 + 4 + 5

= 9 + 16 + 25

= 50 = 7.1 in.

Substitute 3 for l, 4 for w, 5 for h. Simplify.

Next page

CONFIDENTIAL 14

B) The height of a rectangular prism with an 8 ft by 12 ft base and an 18 ft diagonal

= 10.8 ft

18 = 8 + 12 + h 2 22

18 = ( 8 + 12 + h )2 222

2324 = 64 + 144 + h2

h = 116

h = 1162

Substitute 18 for d, 8 for l, 12 for w.Square both sides of the equation.

Simplify.

Solve for h.

Take the square root of both sides.

2

CONFIDENTIAL 15

Now you try!

2) Find the length of the diagonal of a cube with edge length 5 cm.

2) 8.67 cm

CONFIDENTIAL 16

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 17

Graphing Figures in Three Graphing Figures in Three DimensionsDimensions

Graph each 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 18

Graph each figure.

B) A cylinder with radius 3 units, height 5 units, and one base centered at (0, 0, 0)

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

(0, 0, 0)

(0, 0, 5)

(3, 0, 0)

x

z

y(0, 3, 0)

Graph the center of the bottom base at (0, 0, 0).Since the height is 5, graph the center of the top base at (0, 0, 5)The radius is 3, so the bottom base will cross the x-axis at (3, 0, 0) and the y-axis at (0, 3, 0).Draw the top base parallel to the bottom base and connect the bases.

CONFIDENTIAL 19

Now you try!

3) Graph a cone with radius 5 units, height 7 units, and the base centered at (0, 0, 0)

CONFIDENTIAL 20

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 21

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 22

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 23

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

Round to the nearest tenth, if necessary.

B) (3, 8, 10) and (7, 12, 15)

M(x1 + x2)

2,

(y1+ y2)

2,

(z1 + z2)

2

M3 + 7

2,

8 + 12

2,

10 + 15)

2

M(5, 10, 12.5)

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

= (7 - 3)2 + (12 - 8)2 + (15 - 10)2

= 16 + 16 + 25 = 57

= 7.5 units

Distance: Midpoint:

CONFIDENTIAL 24

Now you try!

4a. (0, 9, 5) and (6, 0, 12)

4b. (5, 8, 16) and (12, 16, 20)

4a) 12.89 units ; M(3, 4.5, 8.5)4b) 11.36 units ; M(8.5, 12, 13)

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 25

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 26

Now you try!

5) If both divers swam straight up to the surface, how far apart would they be?

5) units ;()

9 ft

Depth: 8 ft

Depth: 12 ft

18 ft

15 ft

6 ft

CONFIDENTIAL 27

Now some problems for you to practice !

CONFIDENTIAL 28

Assessment

1) Explain why a cylinder is not a polyhedron.

1) A polyhedron is a solid object has vertices, edges and faces which cylinder does not have.

CONFIDENTIAL 29

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

A) B)

A) V= 6, E = 10, F = 66 - 10 + 6 = 22=2

B) V= 6, E = 10, F = 66 - 10 + 6 = 22=2

CONFIDENTIAL 30

3) Find the unknown dimension in each figure. Round to the nearest tenth, if necessary.

A.The length of the diagonal of a 4 ft by 8 ft by 12 ft rectangular prism.

B.The height of rectangular prism with a 6 in. by 10 in. base and 13 in. diagonal

A) 14.97 ftB) 5.74 in.

CONFIDENTIAL 31

4) Graph each figure.

a.A cone with radius 8 units, height 4 units, and the base centered at (0, 0, 0)b.A cylinder with radius 3 units, height 4 units, and one base centered at (0, 0, 0)

CONFIDENTIAL 32

5) Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to nearest tenth, if necessary.

a)(0, 0, 0) and (5, 9, 10)b)(0, 3, 8) and (7, 0, 14)

CONFIDENTIAL 33

6) After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The elevation of the camp is 0.6 km higher than the starting point. What is

the distance from the camp to the starting point?

CONFIDENTIAL 34

Let’s review

Formulas in Three Dimensions 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 35

In the lab before this lesson, you made a conjecture about the relationship between the vertices, edges, and faces of a polyhedron. One

way to state this relationship is given below.

CONFIDENTIAL 36

Euler’s FormulaEuler’s Formula

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

V - E + F = 2.

CONFIDENTIAL 37

Using Euler’s Formula

Find 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 38

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 39

Diagonal of a Right Diagonal of a Right Rectangular PrismRectangular 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 40

Using the Pythagorean Theorem in three Dimensions

Find the unknown dimension in each figure.

A)The length of the diagonal of a 3 in. by 4 in. by 5 in. rectangular prism

2 22d = 3 + 4 + 5

= 9 + 16 + 25

= 50 = 7.1 in.

Substitute 3 for l, 4 for w, 5 for h. Simplify.

Next page

CONFIDENTIAL 41

B) The height of a rectangular prism with an 8 ft by 12 ft base and an 18 ft diagonal

= 10.8 ft

18 = 8 + 12 + h 2 22

18 = ( 8 + 12 + h )2 222

2324 = 64 + 144 + h2

h = 116

h = 1162

Substitute 18 for d, 8 for l, 12 for w.Square both sides of the equation.

Simplify.

Solve for h.

Take the square root of both sides.

2

CONFIDENTIAL 42

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 43

Graphing Figures in Three Graphing Figures in Three DimensionsDimensions

Graph each 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 44

Graph each figure.

B) A cylinder with radius 3 units, height 5 units, and one base centered at (0, 0, 0)

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

(0, 0, 0)

(0, 0, 5)

(3, 0, 0)

x

z

y(0, 3, 0)

Graph the center of the bottom base at (0, 0, 0).Since the height is 5, graph the center of the top base at (0, 0, 5)The radius is 3, so the bottom base will cross the x-axis at (3, 0, 0) and the y-axis at (0, 3, 0).Draw the top base parallel to the bottom base and connect the bases.

CONFIDENTIAL 45

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 46

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 47

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)

Next page

CONFIDENTIAL 48

Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary.

B) (3, 8, 10) and (7, 12, 15)

M(x1 + x2)

2,

(y1+ y2)

2,

(z1 + z2)

2

M3 + 7

2,

8 + 12

2,

10 + 15)

2

M(5, 10, 12.5)

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

= (7 - 3)2 + (12 - 8)2 + (15 - 10)2

= 16 + 16 + 25 = 57

= 7.5 units

Distance: Midpoint:

CONFIDENTIAL 49

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 50

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