Gram–Euler theorem
In geometry, the Gram–Euler theorem generalizes the internal angle sum formula to higher-dimensional polytopes.Statement
Let be an -dimensional convex polytope. For each cell, let be its dimension, and be its internal solid angle, determined by choosing a small enough -sphere centered at some point in the interior of and finding the surface area contained inside. Then,.Examples
For a polygon with sides, there is one face, which has internal angle, and edges, each of which has internal angle. Let be the sum of the internal angles of the corners. The Gram-Euler theorem then tells us that, or equivalently,.
