An n-sided polygon has n sides and n vertices.

Every diagonal and every side of the polygon is the line segment 
connecting a combination of two vertices.  

Every combination of 2 vertices uniquely represents either a side
or a diagonal. 

Derivation of the formula:

1. First we get the number of sides and diagonals by getting the
   number of combinations of n vertices taken 2 at a time:

   C(n,2) =  

2. To get the number of diagonals only, we must subtract the number 
   of sides, which is n. 

    - n

3. We get an LCD of 2 and simplify:

     - 

     = 

    

    

    

So if d represents the number of diagonals of 
an n-sided polygon, then the formula is

d = 

Edwin