CK-12 Geometry
Page 39
The study of probability originated in the seventeenth century as mathematicians analyzed games of chance.
For Further Reading
French mathematicians Pierre de Fermat and Blaise Pascal are credited as the “inventors” of mathematical probability. The reference below is an easy introduction to their ideas. http://mathforum.org/isaac/problems/prob1.html
Review Questions
Rita is retired. For her, every day is a holiday. What is the probability that tomorrow is a holiday for Rita?
Chaz is “on call” any time, any day. He never has a holiday. What is the probability that tomorrow is a holiday for Chaz?
The only things on Ray’s refrigerator door are green magnets and yellow magnets. Ray takes one magnet off without looking. What is the probability that the magnet is green?
What is the probability that the magnet is yellow?
What is the probability that the magnet is purple?
Ray takes off two magnets without looking.
What is the probability that both magnets are green?
What is the probability that Ray takes off one green and one yellow magnet?
Reed uses the diagram below as a model of a highway.
What is the probability that the accident is not between Canton and Dayton?
What is the probability that the accident is closer to Canton than it is to Barton?
Reed got a call about an accident at an unknown location between Acton and Dayton.
A tire has an outer diameter of . Nina noticed a weak spot on the tire. She marked the weak spot with chalk. The chalk mark is along the outer edge of the tire. What is the probability that part of the weak spot is in contact with the ground at any time?
Mike set up a rectangular landing zone that measures by . He marked a circular helicopter pad that measured feet across at its widest in the landing zone. As a test, Mike dropped a package that landed in the landing zone. What is the probability that the package landed outside the helicopter pad?
Fareed made a target for a game. The target is a foot-by-foot square. To win a player must hit a smaller square in the center of the target. If the probability that players who hit the target win is , what is the length of a side of the smaller square?
Amazonia set off on a quest. She followed the paths shown by the arrows in the map.
Every time a path splits, Amazonia takes a new path at random. What is the probability that she ends up in the cave?
Review Answers
, or equivalent
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Approximately
Approximately
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Chapter 11: Surface Area and Volume
The Polyhedron
Learning Objectives
Identify polyhedra.
Understand the properties of polyhedra.
Use Euler’s formula solve problems.
Identify regular (Platonic) polyhedra.
Introduction
In earlier chapters you learned that a polygon is a two-dimensional (planar) figure that is made of three or more points joined together by line segments. Examples of polygons include triangles, quadrilaterals, pentagons, or octagons. In general, an is a polygon with n sides. So a triangle is a , or sided polygon, a pentagon is a , or sided polygon.
You can use polygons to construct a 3-dimensional figure called a polyhedron (plural: polyhedra). A polyhedron is a 3-dimensional figure that is made up of polygon faces. A cube is an example of a polyhedron and its faces are squares (quadrilaterals).
Polyhedron or Not
A polyhedron has the following properties:
It is a dimensional figure.
It is made of polygons and only polygons. Each polygon is called a face of the polyhedron.
Polygon faces join together along segments called edges.
Each edge joins exactly two faces.
Edges meet in points called vertices.
There are no gaps between edges or vertices.
Example 1
Is the figure a polyhedron?
Yes. A figure is a polyhedron if it has all of the properties of a polyhedron. This figure:
Is dimensional.
Is constructed entirely of flat polygons (triangles and rectangles).
Has faces that meet in edges and edges that meet in vertices.
Has no gaps between edges.
Has no non-polygon faces (e.g., curves).
Has no concave faces.
Since the figure has all of the properties of a polyhedron, it is a polyhedron.
Example 2
Is the figure a polyhedron?
No. This figure has faces, edges, and vertices, but all of its surfaces are not flat polygons. Look at the end surface marked A. It is flat, but it has a curved edge so it is not a polygon. Surface B is not flat (planar).
Example 3
Is the figure a polyhedron?
No. The figure is made up of polygons and it has faces, edges, and vertices. But the faces do not fit together—the figure has gaps. The figure also has an overlap that creates a concave surface. For these reasons, the figure is not a polyhedron.
Face, Vertex, Edge, Base
As indicated above, a polyhedron joins faces together along edges, and edges together at vertices. The following statements are true of any polyhedron:
Each edge joins exactly two faces.
Each edge joins exactly two vertices.
To see why this is true, take a look at this prism. Each of its edges joins two faces along a single line segment. Each of its edges includes exactly two vertices.
Let’s count the number of faces, edges, and vertices in a few typical polyhedra. The square pyramid gets its name from its base, which is a square. It has , , and .
Other figures have a different number of faces, edges, and vertices.
If we make a table that summarizes the data from each of the figures we get:
Figure Vertices Faces Edges
Square pyramid
Rectangular prism
Octahedron
Pentagonal prism
Do you see a pattern? Calculate the sum of the number of vertices and edges. Then compare that sum to the number of edges:
Figure
square pyramid
rectangular prism
octahedron
pentagonal prism
Do you see the pattern? The formula that summarizes this relationship is named after mathematician Leonhard Euler. Euler’s formula says, for any polyhedron:
Euler's Formula for Polyhedra
or
You can use Euler’s formula to find the number of edges, faces, or vertices in a polyhedron.
Example 4
Count the number of faces, edges, and vertices in the figure. Does it conform to Euler’s formula?
There are , , and . Using the formula:
So the figure conforms to Euler’s formula.
Example 5
In a faced polyhedron, there are . How many vertices does the polyhedron have?
Use Euler's formula.
There are in the figure.
Example 6
A 3-dimensional figure has , , and . It is a polyhedron? How do you know?
Use Euler's formula.
The equation does not hold so Euler’s formula does not apply to this figure. Since all polyhedra conform to Euler’s formula, this figure must not be a polyhedron.
Regular Polyhedra
Polyhedra can be named and classified in a number of ways—by side, by angle, by base, by number of faces, and so on. Perhaps the most important classification is whether or not a polyhedron is regular or not. You will recall that a regular polygon is a polygon whose sides and angles are all congruent.
A polyhedron is regular if it has the following characteristics:
All faces are the same.
All faces are c
ongruent regular polygons.
The same number of faces meet at every vertex.
The figure has no gaps or holes.
The figure is convex—it has no indentations.
Example 7
Is a cube a regular polyhedron?
All faces of a cube are regular polygons—squares. The cube is convex because it has no indented surfaces. The cube is simple because it has no gaps. Therefore, a cube is a regular polyhedron.
A polyhedron is semi-regular if all of its faces are regular polygons and the same number of faces meet at every vertex.
Semi-regular polyhedra often have two different kinds of faces, both of which are regular polygons.
Prisms with a regular polygon base are one kind of semi-regular polyhedron.
Not all semi-regular polyhedra are prisms. An example of a non-prism is shown below.
Completely irregular polyhedra also exist. They are made of different kinds of regular and irregular polygons.
So now a question arises. Given that a polyhedron is regular if all of its faces are congruent regular polygons, it is convex and contains no gaps or holes. How many regular polyhedra actually exist?
In fact, you may be surprised to learn that only five regular polyhedra can be made. They are known as the Platonic (or noble) solids.
Note that no matter how you try, you can’t construct any other regular polyhedra besides the ones above.
Example 8
How many faces, edges, and vertices does a tetrahedron (see above) have?
Example 9
Which regular polygon does an icosahedron feature?
An equilateral triangle
Review Questions
Identify each of the following three-dimensional figures:
Below is a list of the properties of a polyhedron. Two of the properties are not correct. Find the incorrect ones and correct them. It is a dimensional figure.
Some of its faces are polygons.
Polygon faces join together along segments called edges.
Each edge joins three faces.
There are no gaps between edges and vertices.
Complete the table and verify Euler’s formula for each of the figures in the problem.
Figure # vertices # edges # faces
Pentagonal prism
Rectangular pyramid
Triangular prism
Trapezoidal prism
Review Answers
Identify each of the following three dimensional figures:
pentagonal prism
rectangular pyramid
triangular prism
triangular pyramid
trapezoidal prism
Below is a list of the properties of a polyhedron. Two of the properties are not correct. Find the incorrect ones and correct them. It is a dimensional figure.
Some of its faces are polygons. All of its faces are polygons.
Polygon faces join together along segments called edges.
Each edge joins three faces. Each edge joins two faces.
There are no gaps between edges and vertices.
Complete the table and verify Euler’s formula for each of the figures in the problem.
Figure # vertices # edges # faces
Pentagonal prism
Rectangular pyramid
Triangular prism
Trapezoidal prism
In all cases
Representing Solids
Learning Objectives
Identify isometric, orthographic, cross-sectional views of solids.
Draw isometric, orthographic, cross-sectional views of solids.
Identify, draw, and construct nets for solids.
Introduction
The best way to represent a three-dimensional figure is to use a solid model. Unfortunately, models are sometimes not available. There are four primary ways to represent solids in two dimensions on paper. These are:
An isometric (or perspective) view.
An orthographic or blow-up view.
A cross-sectional view.
A net.
Isometric View
The typical three-dimensional view of a solid is the isometric view. Strictly speaking, an isometric view of a solid does not include perspective. Perspective is the illusion used by artists to make things in the distance look smaller than things nearby by using a vanishing point where parallel lines converge.
The figures below show the difference between an isometric and perspective view of a solid.
As you can see, the perspective view looks more “real” to the eye, but in geometry, isometric representations are useful for measuring and comparing distances.
The isometric view is often shown in a transparent “see-through” form.
Color and shading can also be added to help the eye visualize the solid.
Example 1
Show isometric views of a prism with an equilateral triangle for its base.
Example 2
Show a see-through isometric view of a prism with a hexagon for a base.
Orthographic View
An orthographic projection is a blow-up view of a solid that shows a flat representation of each of the figure’s sides. A good way to see how an orthographic projection works is to construct one. The (non-convex) polyhedron shown has a different projection on every side.
To show the figure in an orthographic view, place it in an imaginary box.
Now project out to each of the walls of the box. Three of the views are shown below.
A more complete orthographic blow-up shows the image of the side on each of the six walls of the box.
The same image looks like this in fold out view.
Example 3
Show an orthographic view of the figure.
First, place the figure in a box.
Now project each of the sides of the figure out to the walls of the box. Three projections are shown.
You can use this image to make a fold-out representation of the same figure.
Cross Section View
Imagine slicing a three-dimensional figure into a series of thin slices. Each slice shows a cross-section view.
The cross section you get depends on the angle at which you slice the figure.
Example 4
What kind of cross section will result from cutting the figure at the angle shown?
Example 5
What kind of cross section will result from cutting the figure at the angle shown?
Example 6
What kind of cross section will result from cutting the figure at the angle shown?
Nets
One final way to represent a solid is to use a net. If you cut out a net you can fold it into a model of a figure. Nets can also be used to analyze a single solid. Here is an example of a net for a cube.
There is more than one way to make a net for a single figure.
However, not all arrangements will create a cube.
Example 7
What kind of figure does the net create? Draw the figure.
The net creates a box-shaped rectangular prism as shown below.
Example 8
What kind of net can you draw to represent the figure shown? Draw the net.
A net for the prism is shown. Other nets are possible.
Multimedia Link The applet here animates how four solids are made from nets. There are two unique nets for the cube and two for the dodecahedron. Unfolding Polyhedra.
Review Questions
Name four different ways to represent solids in two dimensions on paper.
Show an isometric view of a prism with a square base.
Given the following pyramid:
If the pyramid is cut with a plane parallel to the base, what is the cross section?
If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, what is the cross section?
If the pyramid is cut with a plane perpendicular to the base but not through the top vertex, what is the cross section?
&
nbsp; Sketch the shape of the plane surface at the cut of this solid figure.
Cut
Cut
For this figure, what is the cross section?
Draw a net for each of the following:
Review Answers
Name four different ways to represent solids in two dimensions on paper.
Isometric, orthographic, cross sectional, net
Show an isometric view of a prism with a square base.
Given the following pyramid:
If the pyramid is cut with a plane parallel to the base, what is the cross section? square
If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, what is the cross section? triangle