Question

The interior angles of a regular polygon measure ${150}^{\circ }$ each.Then, find the number of diagonals of the polygon.

Answer 1

Each exterior angle$={30}^{\circ }$

$\therefore$ Number of sides$=\frac{{360}^{\circ }}{{30}^{\circ }}$

$=\frac{360\times \frac{\pi }{180}}{30\times \frac{\pi }{180}}=12$

$\therefore$ Number of diagonals${=}^{12}{C}_2-12$

$=\frac{12!}{10!2!}-12$

$=\frac{12\times 11\times 10!}{10!2}-12$

$=6\times 11-12=66-12=54$