Find the number of diagonals of

Question:

Find the number of diagonals of

(i) a hexagon

(ii) a polygon of 16 sides.

Solution:

A polygon of n sides has n vertices. By joining any two vertices we obtain either a side or a diagonal.

$\therefore$ Number of ways of selecting 2 out of $9={ }^{n} C_{2}=\frac{n(n-1)}{2}$

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

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

(i) In a hexagon, there are 6 sides.

$\therefore$ Number of diagonals $=\frac{6(6-3)}{2}=9$

(ii) There are 16 sides.

$\therefore$ Number of diagonals $=\frac{16(16-3)}{2}=104$

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