## angle deficiency

Add up the angles round a vertex for any regular polyhedron... clearly it must be less than 360 degrees otherwise it would be flat! The amount by which it is less than 360 is called the angle deficiency. Look at the dodecahedron net on the right - what is the angle deficiency at each vertex? Sum the total angle deficiency for all vertices on a dodecahedron. Do the same for some other polyhedra... what do you notice?Below is a proof of an amazing link between angle deficiency and the Euler characteristic V - E + F = 2. |