Question

The measures of the interior angles taken in order of a polygon form an arithmetic sequence. The least measurement in the sequence is ${85}^{\circ }$. The greatest measurement is ${215}^{\circ }$. Find the number of sides in the given polygon

Answer 1

Let $n$ denote the number of sides of the polygon.
Now, the measures of interior angles form an arithmetic sequence.
Let the sum of the interior angles of the polygon be
$S_n=a+(a+d)+(a+2d)+....+l$, where $a=85$ and $l=215$.
We have, $S_n=\frac{n}{2}[l+a]...\left(1\right)$
We know that the sum of the interior angles of a polygon is $(n-2)\times {180}^{\circ }$.
Thus, $S_n=(n-2)\times 180$
From $\left(1\right)$, we have $\frac{n}{2}[l+a]=(n-2)\times 180$
$\Rightarrow \frac{n}{2}[215+85]=(n-2)\times 180$
$150n=180(n-2)\Rightarrow n=12$.
Hence, the number of sides of the polygon is $12$.