Diagonals in a Polygon

By

Danielle Habeeb

A diagonal is a line segment that connects non-adjacent vertices in a polygon. Consider the number of diagonals in a triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon. What pattern do you notice? Use this pattern to predict the number of diagonals in a dodecagon.

First, create three segments to form a triangle.  Since you are looking for diagonals you should connect a segment vertex to vertex but the segments cannot be adjacent.  Therefore, the triangle has no diagonal line segments.

Next, create a quadrilateral. Connect each non-adjacent vertex with a line segment.  You should see that there are only two possible diagonal segments.

Then, construct a pentagon.  Again, connect all non-adjacent vertices until all possible diagonal segments are constructed.  You should find that a pentagonal figure should have 5 diagonal segments.

Create a hexagonal figure.  Connect all non-adjacent vertices until all possible diagonals have been created.  When finished you should have 9 diagonal segments.

Create a heptagon.  Connect all vertices until all non-adjacent possible diagonals have been connected.  You should have connected 14 different diagonals.

Create an Octagon.  Connect all vertices until all non-adjacent possible diagonals have been connected.  You should have connected 20 different diagonals.

As I continued to count each of the polygon’s diagonals, I decided to make a chart, listing the number of vertices and the number of diagonals possible.  I noticed that for each polygon there were a different number of diagonals possible for each vertex.  The more sides the figure had the more possible diagonals from each point.  I found that each time you added a side you could also add the additional diagonal from the vertex.  Example, a quadrilateral had 1 diagonal from each vertex, a pentagon had 2 diagonals from each vertex, a hexagon had 3 diagonals from each vertex, etc.   The pattern I used to determine how many diagonals a particular figure could have was that I multiplied the number of vertices by the number of diagonals from each vertex and then halved it.  Example, a hexagon has 6 vertices with 3 diagonals coming from each, so

6 x 3=18, 18/2= 9, the hexagon should have 9 diagonals possible.  It is not 18 because you cannot count the same diagonal twice.

With further exploration, I created the following formula:

n = the number of sides

Therefore, a dodecagon has 54 possible diagonals.