Question

The difference between an exterior angle of $(n-1)$ sided regular polygon and an exterior angle of $(n+2)$ sided regular polygon is $6^{\circ }$ find the value of $n$.

Answer 1

For $(n-1)$ sided regular polygon:
Let measure of each angle is $x$.
Therefore
$$\begin{array}{cc}& (n-1)x=\left(2\right(n-1)-4){90}^{\circ }\\ & x=(n-3/n-1){180}^{\circ }\\ & \text{For}(n+1)\text{sided regular polygon:}\\ & \text{Let measure of each angle is}y\text{.}\\ & \text{Therefore}\\ & (n+2)y=\left(2\right(n+2)-4){90}^{\circ }\\ & y=(n/n+2){180}^{\circ }\end{array}$$
now we have
$$y-x=6^{\circ }$$
$(n/n+2){180}^{\circ }-(n-3/n-1){180}^{\circ }=6^{\circ }$
$(n/n+2)-(n-3/n-1)=1/30$
$30n(n-1)-30(n-3)(n+2)=(n+2)(n-1)$
$$n^2+n-182=0$$
$$(n-13)(n+14)=0$$
$$n=13,-14$$
Thus, the value of $n$ is 13 .