Question

The difference between the exterior angle of a $(n-1)$ sided regular polygon and a $(n+1)$ sided regular polygon is ${15}^{\circ }$. Find the value of $n$.

Answer 1

Exterior angle of a $n$ sided regular polygon $=\frac{{360}^{\circ }}{n}$

According to the question,

$\frac{{360}^{\circ }}{n-1}-\frac{{360}^{\circ }}{n+1}={15}^{\circ }$

$\frac{1}{n-1}-\frac{1}{n+1}=\frac{{15}^{\circ }}{{360}^{\circ }}$

$\frac{n+1-n+1}{(n-1)(n+1)}=\frac{1}{24}$

$\frac{2}{n^2-1}=\frac{1}{24}$

$n^2-1=24\times 2$

$n^2=49=7^2$

$\therefore n=7$ ($n$ cannot be negative)