Spherical excess formula
= Spherical excess formula
A geodesic triangle on the unit sphere with interior angles $\alpha,\beta,\gamma$ has area
$$
\alpha+\beta+\gamma-\pi.
$$
For a sphere of radius $R$, multiply by $R^2$. Triangulation gives a convex spherical $n$-gon area $\sum_i\alpha_i-(n-2)\pi$.