Diagonals of Polygons

Date: 05/21/97 at 15:36:00
From: pat kelly
Subject: Formulas to find diagonals

My teacher showed us how to find diagonals using two different 
equations. I lost the sheet which I wrote them on. Do you know them? 
One of them might have been n(n-3)/2 and I am not sure of the other 
one, but it was called recursive. 

Date: 05/21/97 at 17:17:13
From: Doctor Wilkinson
Subject: Re: Formulas to find diagonals

The question apparently is "How many diagonals does a polygon with n 
sides have?"  

You have remembered the first formula correctly: it is n(n-3)/2.  
One way to see this is to notice that you can draw (n-3) diagonals 
from every vertex of the polygon. This is because there are (n-1) 
other vertices, but two of them are adjacent vertices and so 
don't count towards making diagonals. This seems to give n(n-2)
diagonals, but this way of counting counts every diagonal twice since 
each diagonal connects to two vertices, so you have to divide by 2.

By a recursive formula, we mean a way of expressing the answer for n
vertices in terms of the answer for n-1 vertices. Suppose, for
example, that you already know the answer for a polygon with n-1
vertices. Now if you add another vertex between two of the vertices
of the original polygon, then all the diagonals of the original 
polygon will still be diagonals of the new polygon, and so will the 
side joining the two vertices that you added a new vertex between, and 
so will the line segments joining the new vertex to all the other 
vertices of the original polygon. (Got all that?) So if we let diag(n) 
be the number of diagonals for a polygon with n sides, we get the 
formula:

        diag(n) = diag(n-1) + n - 3 + 1 or

                  diag(n-1) + n - 2

The first formula is better, since it actually gives you the answer.  
But sometimes it's easier to get a recursive formula first and use 
that to get an explicit formula (your first formula is an explicit one 
since you only need the number of vertices in the polygon to get the 
number of diagonals in that polygon).  This is called "solving the 
recursion."  Sometimes a recursive formula is the best you can do 
because there simply is no explicit formula.

I hope this helps.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/