Question

If the difference between the exterior angle of a $n$ sided regular polygon and an $(n+1)$ sided regular polygon is ${12}^{\circ }$, find the value of $n$.

• A. $n=6$
• B. $n=5$
• C. $n=9$
• D. $n=12$

Answer 1

Answer: (B)

$n=5$

When number of sides of a regular polygon = $n$,
The value of its each exterior angle = $\frac{{360}^{\circ }}{n}$
When number of sides of a regular polygon = $n+1$,
The value of its each exterior angle = $\frac{{360}^{\circ }}{n+1}$
Given $\frac{{360}^{\circ }}{n}-\frac{{360}^{\circ }}{n+1}={12}^{\circ }$

$\Rightarrow {360}^{\circ }(n+1)-{360}^{\circ }\left(n\right)={12}^{\circ }(n+1)n$
$\Rightarrow {360}^{\circ }[n+1-n]={12}^{\circ }(n^2+n)$
$\Rightarrow {360}^{\circ }={12}^{\circ }(n^2+n)$
$\Rightarrow {30}^{\circ }=n^2+n$
$\Rightarrow n^2+n-30=0$
$\Rightarrow (n+6)(n-5)=0$
$\Rightarrow n=5$ or $n=-6$
Since, $n$ can't be negative.
$\therefore n=5$