The inradius of an icosahedron

HOME

Math Central

Quandaries & Queries

Question from fnavroth:

Suppose you have an equilateral triangle.
The area of the triangle is exactly 1200 square centimetres.
Now suppose you have twenty of those triangles.
It's possible to assemble those twenty triangles into a closed
three-dimensional shape, a regular polyhedron.
What would be the volume, in cubic centimetres, of the largest sphere
that could fit inside the shape?

Fnavroth,

It is easy to compute the inradius of an icosahedron, but it is even easier to look it up. I use the appendix of Coxeter's REGULAR POLYTOPES, but you can find it on the web:

http://en.wikipedia.org/wiki/Icosahedron

Of course, you also need the formula for the area of an equilateral triangle and for the volume of a sphere.

Chris

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.