The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

Abstract

We calculate E[V 4(C)], the expected volume of a tetrahedron whose vertices are chosen randomly (i.e. independently and uniformly) in the interior of C, a cube of unit volume. We find $$E[V_4 {\rm (cube)}]={3977\over 216000} - {\pi^2 \over 2160}=0.01384277574\ldots$$ The result is in convincing agreement with a simulation of 3000 · 106 trials.