What is the easiest way to see that, in a regular polyhedron, the sum of the face angles about each vertex is less than 2π? This seems elementary, but I truly have very little background in geometry. Preliminary searches on PF and Google/Wikipedia didn't turn up anything substantial aside from hints that there are more general results regarding angle defects. I am looking for both some intuitive reasoning and some rigor, if possible. Please go slowly, as I am completely lacking in 3D visualization and geometric rigor. Thank you all in advance!