Counting Sides by Counting Diagonals

Date: 06/04/2002 at 07:41:53
From: Sunita
Subject: Sides in a Polygon

I wish to have a formula that can tell me how to calculate the number
of sides of a polygon if I know the number of diagonals in that 
polygon. For example, I know that the number of diagonals in a 
polygon is 44. How many sides does this polygon have?

Date: 06/04/2002 at 07:48:45
From: Doctor Ian
Subject: Re: Sides in a Polygon

Hi Sunita,

Suppose you have a polygon with N vertices.  From each vertex, 
you can draw a diagonal to (N-3) other vertices, so the number of 
directed diagonals is N(N-3).  

However, when you do this, you end up with diagonals going in 
both directions, so you want to cut the number in half.  So the 
number of undirected diagonals is N(N-3)/2.  Convince yourself 
that this works for any N. 

To go from the number of diagonals to the number of vertices, you 
just go in the other direction:

  44 = N(N-3)/2

  88 = N(N-3)

  11 = N

So the polygon has 11 sides. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/