hyperbolic quadrilateral angles

Question

On the hyperbolic plane, if I have a quadrilateral that has all congruent interior angles $\alpha$, how do I figure out what $\alpha$ is? I know in Euclidean geometry one could just use $\frac{180(n-2)}{n}$.

Answer 1

This can be any $\alpha<\displaystyle\frac{180^\circ(n-2)}n$. (For any such $\alpha$ there exists an $n$-gon with equal sides.)

For illustration, pick a regular $n$-gon, and zoom its vertices out from the center of the $n$-gon (equally raise the distance on the rays from the center), then connect again the new $n$ vertices. You will get a regular $n$-gon with bigger area and smaller angles.