My favorite theorem is the General Stokes' Theorem. Basically said, this uses the boundary of an object to find the area or volume thereof. This states:
Let M be an oriented k-manifold in R^{3} (k = 2 or 3) contained in some open set K. Suppose $ \omega $ is (k-1)-form on K. Then:
$ \int _{\it \partial M} \omega\ = {\displaystyle \int _{\it M}\,d\omega} $.
