Question

How many diagonals are there in a polygon with $n$ sides?

Answer 1

A polygon of n sides has n vertices. By joining any two vertices of a polygon, we obtain either a side or a diagonal of the polygon. Number of line segments obtained by joining the vertices of an n sided polygon taken two at a time

= Number of ways of selecting 2 out of n$=n_{C_2}=\frac{n(n-1)}{2}$

Out of these lines, n lines are the sides of the polygon.

Therefore, Number of diagonals of the polygon = $\frac{n(n-1)}{2}-n=\frac{n(n-3)}{2}$.