Tim's Challenge: The angles between faces of a dodecahedron

The dodecahedron is one of the 5 platonic solids. It is a regular convex polyhedra with each of its 12 sides being a regular pentagon. Using trigonometry and/or vector geometry, can you find the angle between two adjacent sides of the dodecahedron?
To help, the applet below attempts to show a method of constructing a dodecahedron around a cube. Click "Draw cube" and then "Draw dodecahedron" or the other way around.

A roof like structure is formed on each face of the cube. The roof itself is constructed from the pentagons as shown in the following diagrams:

The angle between adjacent faces can then be found by finding the lengths h and p.

angle = 2 cos-1 (.5/sin72)
= 116.565

For a brief introduction to the five Platonic polyhedra with a much better applet which also illustrates dual shapes and other inclusions, try this link to Gian Marco Todesco's page.