The area formula for hyperbolic triangles

Abstract

We prove an area formula in terms of side lengths for hyperbolic triangles. The proof is analogous to a proof given by Leonhard Euler in the spherical case. We take this opportunity to prove other results in hyperbolic geometry using the variational techniques that Euler introduced in his work on spherical geometry. We consider in particular the famous Lexell Problem, that is, the problem of finding the locus of vertices of triangles of fixed area and fixed basis.