A hyperbolic geometry is a non-Euclidian geometry with constant negative curvature. It has the property that given a line and a point, many lines can be drawn containing the point that never meet the given line. The picture below (Circle Limit by M.C.Escher) is a conformal map of hyperbolic geometry to Euclidian plane.
How do you show that in a space with hyperbolic geometry, area of a triangle with angles $\alpha$, $\beta$, and $\gamma$ is $\Delta\propto\pi-\alpha-\beta-\gamma$?
A similar result exists for sphere: $\Delta=R^2(\alpha+\beta+\gamma-\pi)$. I could prove this, but I suspect my proof cannot be modified for hyperbolic geometry. My proof for the sphere is fairly trivial, it makes use of symmetry and a Venn diagram.