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 ℓ^__(V = 1) | = | 3^(1/3)(sqrt(2)/7 - (37π)/315 + 4/15 tan^(-1)(sqrt(2)) + (113ln3)/(210sqrt(2))) | = | 0.729462... (OEIS A366019; Beck 2023).

mean line segment length | regular tetrahedron | tetrahedron tetrahedron picking | tetrahedron triangle picking

PolyhedronData