# If sum of all the interior angles of a polygon is 2340^@, how many sides does it have?

Feb 24, 2017

$s = 15$

#### Explanation:

Whatever the number of sides of a polygon,

the sum of its exterior angles is always ${360}^{\circ}$

Further, each pair of exterior angle and interior angle adds up to ${180}^{\circ}$

Hence in a polygon with $s$ sides (or angles),

the sum of all the interior and exterior angles would be ${180}^{\circ} \times s$

and sum of interior angles would be ${180}^{\circ} \times n - {360}^{\circ} = {180}^{\circ} \left(s - 2\right)$

As sum of angles is ${2340}^{\circ}$

Hence, $180 \left(s - 2\right) = 2340$ or $s - 2 = \frac{2340}{180} = 13$

and $s = 13 + 2 = 15$ and polygon is a Pentadecagon.