Hyperbolic triangle

Summary

A hyperbolic triangle is a triangle in the hyperbolic plane: three points not on one line, joined by geodesic segments. Its angle sum is always less than $\pi$, and in curvature $-1$ its area equals the defect $\pi - (\alpha + \beta + \gamma)$. So no hyperbolic triangle has area more than $\pi$. Unlike in the Euclidean plane, a hyperbolic triangle is determined up to isometry by its three angles. Its sides and angles are related by the hyperbolic law of cosines.