Question

If the difference between an exterior angle of regular polygon of $n$ sided regular polygon and $n+1$ sided regular polygon is is $4^{\circ }$. Find $n$.

Answer 1

Sum of exterior angles of a polygon$={360}^{\circ }$

An $n$ sided polygon has $n$ equal exterior angles.

$\frac{{360}^{\circ }}{n}-\frac{{360}^{\circ }}{n+1}=4^{\circ }$

$\Rightarrow {360}^{\circ }[\frac{1}{n}-\frac{1}{n+1}]=4^{\circ }$

$\Rightarrow {90}^{\circ }\left[\frac{n+1-n}{n(n+1)}\right]=1$

$\Rightarrow n^2+n-90$

$\Rightarrow n^2+10n-9n-90=0$

$\Rightarrow n(n+10)-9(n+10)=0$

$\Rightarrow n=9,-10$

$\therefore n=9$