Suppose you have a sphere of radius R and you have four planes that are all tangent to the sphere such that...

geometry/tetrahedron

Problem

Suppose you have a sphere of radius R and you have four planes that are
all tangent to the sphere such that they form an arbitrary tetrahedron
(it can be irregular).  What is the ratio of the surface area of the
tetrahedron to its volume?

Show solution

For each face of the tetrahedron, construct a new tetrahedron with that
face as the base and the center of the sphere as the fourth vertex.
Now the original tetrahedron is divided into four smaller ones, each of
height R.  The volume of a tetrahedron is Ah/3 where A is the area of
the base and h the height; in this case h=R.  Combine the four
tetrahedra algebraically to find that the volume of the original
tetrahedron is R/3 times its surface area.