Polygons with two different angles

Question

I have the following questions:

Let us consider polygons with only two different interior angles $\alpha$ and $\beta = \pi - \alpha$.

For which frequencies of these angles is it possible to form a closed polygon?

Answer 1

If the interior angles of a simple polygon are $\alpha_1,\ldots,\alpha_n$, then $\sum(\pi-\alpha_i)=2\pi$. Here we have $k_1$ times $\alpha$ and $k_2$ times $\beta$, so need $k_1(\pi-\alpha)+k_2(\pi-\beta)=2\pi$, or $k_1\pi+(k_2-k_1)\alpha=2\pi$, which amounts to either $k_1=k_2=2$ or $$\alpha=\frac{(k_1-2)\pi}{k_1-k_2}. $$