Question

The difference between the exterior angles of two regular polygons, having the sides equal to $(n-1)$ and $(n+1)$ is $9^{\circ }$. Find the value of $n$.

• A. $n=11$
• B. $n=12$
• C. $n=8$
• D. $n=9$

Answer 1

The correct option is D $n=9$

When number of sides of a regular polygon $=$ $n-1$,
The value of its each exterior angle $=$ $\frac{{360}^{\circ }}{n-1}$
When number of sides of a regular polygon $=$ $n+1$,
The value of its each exterior angle $=$ $\frac{{360}^{\circ }}{n+1}$
Given, $\frac{{360}^{\circ }}{n-1}-\frac{{360}^{\circ }}{n+1}=9^{\circ }$
$\Rightarrow {360}^{\circ }(n+1)-{360}^{\circ }(n-1)=9^{\circ }(n+1)(n-1)$
$\Rightarrow {360}^{\circ }[n+1-(n-1\left)\right]=9^{\circ }(n+1)(n-1)$
$\Rightarrow {360}^{\circ }\left(2\right)=9^{\circ }(n^2-1)$
$\Rightarrow {40}^{\circ }\left(2\right)=n^2-1$
$\Rightarrow {80}^{\circ }=n^2-1$
$\Rightarrow n^2=81$
$\Rightarrow n=9$