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 othervertices. This process can be repeated for all n vertices to givevertices altogether. Since however, a line drawn from vertexto vertexonly retraces the line drawn from vertexto vertex we must divide by two so that diagonals are not duplicated.

There arediagonals altogether.

A proof by induction is also possible.

Letbe the statement 'a polygon withsides hasdiagonals'. If(a triangle) there arediagonals, sois true.

Supposeis true, so that a polygon withsides hasdiagonals. If an extra vertex is added, we can drawlines from this 'extra' vertex to the others, and one side becomes a diagonal. There will be

Henceis true and the statement is proved.