Question

Find the number of sides of a regular polygon if each interior angle is ${135}^o$.

Answer 1

Let $\hat{i}$ and $\hat{e}$ denote the interior and exterior angles of a regular $n$-gon.
We know that $n\hat{e}={360}^o$ and $\hat{i}+\hat{e}={180}^o$.
We are given that $\hat{i}={135}^o$. Hence
$\hat{e}={180}^o-{135}^o={45}^o$.
This gives
$n=\frac{{360}^o}{\hat{e}}=\frac{{360}^o}{{45}^o}=8$.
Thus the number of sides is 8.