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Joe Joyner's Homepage - Norview High School

Joe Joyner
Mathematics Teacher
Norview High School
Norfolk, VA
(757) 441-5865

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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
3 0

    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 see formula)

        If you wish to see a justification of the formula, send email to me at Norview High School.

Created by Joe Joyner 10/17/99
Revised by Dr. Marcia Tharp 11/30/99