Gauss-Bonnet Formula -- from Wolfram MathWorld

The Gauss-Bonnet formula has several formulations. The simplest one expresses the total Gaussian curvature of an embedded triangle in terms of the total geodesic curvature of the boundary and the jump angles at the corners. More specifically, if M is any two-dimensional Riemannian manifold (like a surface in three-space) and if T is an embedded triangle, then the Gauss-Bonnet formula states that the integral over the whole triangle of the Gaussian curvature with respect to area is given by...