Create a tetrahedron starting with a square ABCD (clockwise), add 2 points E and F placing them halfway between AB and AD respectively.

Fold along lines EF, EC and FC.

Then, form the tetrahedron by keeping triangle EFC as the base, and folding triangles AEF, BEC, and DFC up so that points A, B, and D all meet.

For AB=a find the radius of the largest sphere contained by this tetrahedron

For a tetrahedron we have two formulas for the volume

Volume = (Area of Base) x Height / 3 and

Volume = (Surface Area) x Inradius / 3.  Hence

Inradius = (Area of Base) x Height / (Surface Area).

Initially take AB=1 and use AEF as base. Then Area of Base = 1/8, Height = 1, and Surface Area = 1 so that Inradius = 1/8. Multiplying up by a then gives the general result

Inradius = a/8.

Posted by Richard on 2005-04-20 06:23:09