Given a regular tetrahedron of unit volume, consider the lengths of line segments connecting pairs of points picked at random
inside the tetrahedron. The distribution of lengths is illustrated above and the
mean line segment length can be given in
closed form as

This beautiful result supplants the approximate value estimated using quasi-Monte Carlo
numerical integration by E. Weisstein in Feb. 2005. (In fact, numerical
integration using a global adaptive method with maximum error increases gives a much more accurate estimate
of 0.729462.)

To obtain the mean line segment length for a regular tetrahedron with unit edge lengths (instead of unit volume), solve (where is the volume of a tetrahedron in terms of its edge length)
for
to obtain
and take
to give

Beck, D. "Mean Distance in Polyhedra." 22 Sep 2023. https://arxiv.org/abs/2309.13177.Sloane,
N. J. A. Sequence A366019 in "The
On-Line Encyclopedia of Integer Sequences."