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

Once you make the tetrahedron, if you set it up on the smallest face (making triangle AEF the base), then you end up with a base of 1 x 1 x ��2 (where 1 = 1/2 a).

Two lateral faces will form 90�� angles with the base (DFC and BEC) and these triangular faces have dimensions 1, 2, ��5 (where 1 = 1/2 a).

Obviously, the sphere will touch each face, and I can easily see the sphere touching faces AEF, BEC, and DFC (they form a right corner), but I can't figure out how large the sphere is when it touches EFC.

Posted by Erik O. on 2005-04-19 18:43:11