## Convex Polygon

A convex polygon can be defined as a polygon with all its interior angles less than 180 degrees. Also, all diagonals lie entirely inside a convex polygon. If a convex polygon has 324 diagonals, how many sides does this polygon have?
Source: mathcontest.olemiss.edu 12/10/2007

SOLUTION

A triangle has 0 diagonal.

A pentagon has 5 diagonals.

General formula
Consider a polygon with $n$ sides. It has $n$ vertices.

Pretend you are standing at a vertex called “You” in the figure above and facing the interior of the convex polygon. How do you know which way is the interior? If you see no vertices in front of you, you are facing the exterior of the polygon. If you see vertices you are facing the interior of the polygon. You have two neighbors one on your immediate left and one on your immediate right. You also have a lot of non-neighboring vertices to which you are connected by diagonals. In fact, you have exactly $n-1-2=n-3$ non-neighboring vertices to which you are connected by $n-3$ diagonals. We subtract 1 for yourself and 2 for your 2 neighbors.

What is true about you is also true about each of the $n$ vertices of the polygon. So, the polygon has a total of $\left (n-3\right )n\div 2$ diagonals. We divide by 2 because we double count the diagonals.

Check the formula
Triangle: $\left (3-3\right )3\div 2=0$ diagonal
Quadrilateral: $\left (4-3\right )4\div 2=2$ diagonals
Pentagon: $\left (5-3\right )5\div 2=5$ diagonals
Hexagon: $\left (6-3\right )6\div 2=9$ diagonals

Let’s draw a hexagon to verify that there are 9 diagonals:

324 diagonals
We want to know how many sides are in a polygon with 324 diagonals. We write the following equation:
$324=\left (n-3\right )n\div 2$
$2\times 324=\left (n-3\right )n$
$648=\left (n-3\right )n$

Let’s try a few numbers for $n$:
$22\times 25=550$
$23\times 26=598$
$24\times 27=648$

So, the polygon has 27 sides.