Question

If the difference between an exterior angle of an (n - 1) sided regular polygon and an (n + 1) sided regular polygon is $9^{\circ }$, the find the value of ‘n’.

Answer 1

Each exterior angle of an (n - 1) sided regular polygon

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

Each exterior angle of an (n + 1) sided regular polygon

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

Given:
$\frac{{360}^{\circ }}{n-1}-\frac{{360}^{\circ }}{n+1}=9^{\circ }$

$\Rightarrow 360°(\frac{1}{n-1}-\frac{1}{n+1})=9^{\circ }$
$\Rightarrow 40\left(\frac{n+1-(n-1)}{(n-1)(n+1)}\right)=1$
$\Rightarrow 40(n+1-n+1)=n^2-1$
$\Rightarrow 80=n^2-1$
$\Rightarrow n^2=81$
$\Rightarrow n=9or-9$
As number of sides is positive, n = 9