Question

The interior angles of a convex polygon are in Arithmetic Progression. The smallest angle is ${120}^{\circ }$ and the common difference is $5^{\circ }$. What is the number of sides of the polygon?

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Let the number of sides be ‘$n$’
Smallest angle = ${120}^{\circ }$ , common difference =$5^{\circ }$.
Sum of interior angles of the convex polygon is given as, = ${180}^{\circ }$ $\times (n-2)$
i.e., $\left(\frac{n}{2}\right)(2\times {120}^{\circ }+5\times (n-1\left)\right)$ = ${180}^{\circ }\times (n-2)$
$(n\times {240}^{\circ }+5n^2-5n)=(n\times {360}^{\circ })-{720}^{\circ }$
$5n^2-125n+720=0$
$n=9$