Question

If the interior angles of a polygon are in A.P. with common difference $5^{\circ }$ and the smallest angle is ${117.5}^{\circ },$ then the number of sides of the polygon is

• A. $8$
• B. $18$
• C. $8,18$
• D. $8,12$

Answer 1

The correct option is A $8$

If the interior angles of a polygon are ${117.5}^{\circ },{122.5}^{\circ },{127.5}^{\circ },...,$ then their corresponding exterior angles are ${62.5}^{\circ },{57.5}^{\circ },{52.5}^{\circ },...$
We know that sum of exterior angles is equal to ${360}^{\circ }$
$\Rightarrow 360=\frac{n}{2}[2\times 62.5+(n-1\left)\right(-5\left)\right]$
$\Rightarrow 720=n[125-5n+5]$
$\Rightarrow n^2-26n+144=0$
$\Rightarrow n=8$ or $n=18$
If $n=18,$ then some exterior angles become negative.
$\therefore n=8$