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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.

##### Answer:

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.

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