Hyperbolic hexagon

A six-sided polygon in the hyperbolic plane whose sides are geodesic segments. The especially useful right-angled case is a right-angled hyperbolic hexagon.

A geodesic hexagon in the hyperbolic plane with all six interior angles $\pi /2$. Any three positive alternating side lengths determine it up to isometry. If those sides are ${\ell }_i/2$, the opposite seam satisfies $\cosh s_i=(\cosh ({\ell }_i/2)+\cosh ({\ell }_j/2)\cosh ({\ell }_k/2))/(\sinh ({\ell }_j/2)\sinh ({\ell }_k/2))$. Two congruent such hexagons glued along the other three sides form a pair of pants with boundary lengths ${\ell }_1,{\ell }_2,{\ell }_3$.