jake baker’s maths blog

I spent quite some time over the Easter holidays finding expressions for the volume of a dodecahedron and icosahedron (with edge lengths 1 unit).  I found that there is a known and perhaps surprising result, that a dodecahedron fills more of a sphere which it just touches (a circumscribing sphere) than an icosahedron.

 Here are the results:

The interesting parts of the calculations were finding the angles between the faces of the polyhedra, and using various trigonometric identities to express the results as surds, leading to the really quite simple expressions for the volumes and radii.

One key result is that cos72 = (-1+sqrt(5))/4.  From this fact you can find the trig functions of lots of other angles.

Next I’ll find some images of a dodecahedron and icosahedron, though of course there are beautiful rotating ones on http://mathworld.wolfram.com/topics/PlatonicSolids.html.