Spherical Geometry - Main Theorem

TheoremGirard's Theorem

The area of a spherical triangle with interior angles $\alpha$ , $\beta$ , $\gamma$ on a sphere of radius $R$ is:

A = R^2(\alpha + \beta + \gamma - \pi)

The quantity $E = \alpha + \beta + \gamma - \pi$ is called the spherical excess.

This fundamental result, discovered by Albert Girard (1629), establishes that spherical triangles encode their area in angular data. Unlike Euclidean geometry where all triangles have angle sum $\pi$ regardless of size, spherical triangles with larger angle sums have proportionally larger areas. The spherical excess directly measures area normalized by $R^2$ .