Diagonals in a polygon

By

Tennille Rushin

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 (12-sided polygon).

First create a triangle.   Since you are looking for diagonals you must go from vertex to vertex, but they cannot be adjacent to each other.  Therefore a triangle has no diagonals.

Next create a quadrilateral.  Create segments from vertex to vertex.  A quadrilateral has two diagonals that are not adjacent to each other.

Construct a pentagon.  Once all the diagonals have been connected you will have 5 possible diagonals.

Once you’ve created a hexagon, create diagonals from each vertex.  You will see that there are 9 diagonals.

Create a heptagon to figure out a pattern.  There are 14 diagonals in the heptagon.

As I looked at each polygon I tried to find a relationship between the number of vertices and diagonals in the polygon.  At first I compared sides and diagonals and made a tentative conjecture.  When I finished the hexagon, (which had 8 diagonals), I saw a pattern and thought the next polygon would have 11 sides.  Well the heptagon ended up having 12, so I knew that wasn’t right.  I started looking at the number of vertices and comparing it to the diagonals.  The heptagon has 7 vertices and 4 diagonals coming from each.  This means 28 diagonals when you multiply them together, but you have to account for the overlapping.  So you take half of the total, meaning 14 diagonals.  I checked to make sure this process worked with the other polygons.  The hexagon had 6 vertices with 3 diagonals which gives you 18 divided by 2 equals 9 diagonals.  I tested the others and this method worked.  I also noticed a pattern between the number of sides and how many diagonals came from each vertex.  The pentagon had 2, hexagon 3, heptagon 4, and if the pattern continues the octagon has 5.

With further thought and conversation with other colleagues I developed a formula that can

be used at all times.  This formula lets you calculate how many diagonals you would have for any polygon.

Without constructing a dodecagon we can calculate the number of diagonals, 54.