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I am trying to get a relation between the length of the sides and the angles of a hyperbolic pentagon. In literature I can find relations for pentagons which has at least three Right angle. So my question is the following.

Let S be a hyperbolic pentagon with angles $\alpha_1,...,\alpha_5$ and with sides $l_1,...,l_5$ such that $\alpha_i\neq\pi/2$ for all $i$. Is there any formula relating $\alpha_i$ with $l_i$.

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up vote 3 down vote accepted

No, there is not, and yes there is. Since the side lengths do not determine the angles. To be precise: there is a ten-dimensional family of quintuples of points in the hyperbolic plane, and a three-dimensional isometry group, so modding out by that, there is a two-dimensional family of pentagons with prescribed side lengths (there is a dual argument for when we know the angles). This means that on the one hand, we cannot determine the angles from the sides, and on the other hand, two angles determine the rest. In each of the combinatorial possibilities for the pair of angles, you can solve the various cosine laws to determine the others, which is instructive, but not that much fun.

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