Tetrahedron line picking considers the lengths of line segments connecting two points picked independently and uniformly with respect to volume inside a regular tetrahedron of unit volume. The statistical distribution of lengths is illustrated above, and the mean line segment length can be given in closed form as
 |  |  |
(1)
|
 |  |  |
(2)
|
(OEIS A366019; Beck 2023, p. 15).
This result supplants the approximate value 
estimated using quasi-Monte
Carlo 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), use the volume 
of a regular tetrahedron with edge length
.
Unit volume gives 
, so divide 
by this value to give
 |  |  |
(3)
|
 |  |  |
(4)
|
This value is implemented in the Wolfram Language as PolyhedronData["Tetrahedron", "MeanInteriorLineSegmentLength"].
