Question

Find the number of diagonals of (i) a hexagon (ii) a polygon of 16 sides.

Answer 1

A polygon of n sides has n vertices. By joining any two vertices we obtain either a side or a diagonal.
∴ Number of ways of selecting 2 out of 9 ${=}^n{C}_2=\frac{n\left(n-1\right)}{2}$
Out of these lines, n lines are the sides of the polygon.

∴ Number of diagonals = $\frac{n\left(n-1\right)}{2}-n=\frac{n\left(n-3\right)}{2}$

(i) In a hexagon, there are 6 sides.
∴ Number of diagonals = $\frac{6\left(6-3\right)}{2}=9$

(ii) There are 16 sides.
∴ Number of diagonals = $\frac{16\left(16-3\right)}{2}=104$