## Proof That a Convex n – Sided Polygon Has n(n-3)/2 Diagonals

A diagonal is a straght line in the interior of a polygon that goes from one vertex to another.

A triangle has no diagonals, while a square has two and a pentagon has six. Each vertex is connected by edges of the polygon to two other vertices, so straight lines draw form the first vertex to the other two cannot be interior to the polygon and wont be diagonals. If there are n vertices altogether, a straight line can be drawn from the first vertex to the other vertices. This process can be repeated for all n vertices to give vertices altogether. Since however, a line drawn from vertex to vertex only retraces the line drawn from vertex to vertex we must divide by two so that diagonals are not duplicated.

There are diagonals altogether.

A proof by induction is also possible.

Let be the statement 'a polygon with sides has diagonals'. If (a triangle) there are diagonals, so is true.

Suppose is true, so that a polygon with sides has diagonals. If an extra vertex is added, we can draw lines from this 'extra' vertex to the others, and one side becomes a diagonal. There will be Hence is true and the statement is proved. 