Imagine a "flying saucer" obtained by gluing together two discs along their boundaries. The object you obtain is a kind of singular surface that is flat everywhere except on the glued boundary where it should have some sort of "curvature". But how would you define such curvature? In this talk I will show how to use the Gauss-Bonnet theorem to give a definition for this and for many other examples that leads to a simple computable formula. This will be an elementary talk in which all concepts will be carefully defined. My goal is to convince you that, if this is not already well-known, it should be.