Question

Is it possible to have a regular polygon whose each interior angle is :
${138}^o$

Answer 1

Suppose polygon has $n$ side

The sum of interior angles

$=(n-2)\times {180}^o$

each exterior angles is $=\frac{(n-2)\times 180}{n}$

So $n\times {45}^o$

$\therefore 180n-360=45n$

$135n=360orn=\frac{{360}^o}{135}=\frac{8}{3}$

As ${}^{\prime }{n}^{\prime }$ should be $+ve$ integrate, so it is not possible to have such a polygon.