Let T be a possibly irregular tetrahedron, let V be its volume, let S be
its surface area, and let R be the radius of the largest sphere inscribed
in T. Express the ratio V/S in terms of R.
Let the four faces of T have areas A1, A2,
A3 and A4. Then:
S = A1 + A2 + A3 + A4
Now, decompose T into four smaller tetrahedra, each one having a face of T
as base, and the center of the inscribed sphere as the opposite point.
The height of each of these tetrahedra is R, so:
| V | = A1R/3 + A2R/3 + A3R/3 + A4R/3 |
| | = SR/3 |
So...
V/S = R/3