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Explorations in the Regular Polygon
A polygon is
a closed, plane figure that is composed of line segments that are connected
to each other only at their endpoints and no endpoint is unconnected.
What is the simplest type of polygon that you can draw? (click
to see answer)
The endpoints of the line segments that compose the
polygon are also called vertices of the polygon. Two vertices
that lie on the same line segment are called consecutive vertices.
Two vertices that do not lie on the same line segment are nonconsecutive.
A diagonal of
a polygon is a line segment that connects two non-consecutive vertices.
Using the triangle that you just drew, how many diagonals
can be drawn in the polygon?
The next simplest type of polygon will have how many
sides? Draw an example of it.
Now draw all of the possible diagonals that you can
in this polygon. How many are there?
For a triangle (n = 3 sides), there
were no diagonals.
(Click here to see example)
For a quadrilateral (n = 4 sides),
there were 2 diagonals. (Click here to see example)
Let's keep track of our information by creating a table of the results.
Diagonals of the Polygon
|Number of Sides (n)
||Number of Diagonals
Draw a polygon that has 5 sides, another with 6 sides
and another with 7 sides. For each polygon, draw all of the possible
diagonals and record your results on a table similar to the one above.
Done? Did drawing all of the diagonals get
to be rather difficult as you got to polygons with more and more sides?
What was difficult about it? Double check your work.
For a pentagon (n = 5 sides), there
were 5 diagonals. (Click here to see example)
For a hexagon (n = 6 sides), there
were 9 diagonals. (Click here to see example)
For a heptagon (n = 7 sides), there
were 14 diagonals. (Click here to see example)
Now let's see if we can find
a pattern to the number of diagonals that we drew in our experiment.
Why look for a pattern? If we can find a pattern, then we can
use it to predict how many diagonals there are in a polygon.
In that way, we won't have to draw the polygon and its diagonals and then
count the number of diagonals. Imagine how difficult that would be
for a polygon with 20 sides!
Review the numbers that you now have entered in your
copy of the Diagonals of the Polygon table above. Do you see a pattern?
Can you support your answer. (Click here to
If you wish to see a justification
of the formula, send email to me at Norview
Created by Joe Joyner 10/17/99
Revised by Dr. Marcia Tharp 11/30/99