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Mean Tetrahedron Volume

The mean tetrahedron volume V^_ is the average volume of a tetrahedron in tetrahedron picking within some given shape. As summarized in the following table, it is possible to compute the mean tetrahedron volume in closed form for tetrahedron picking for some simple shapes.

shapeV^_
ball tetrahedron picking(12pi)/(715)
cube tetrahedron picking(3799)/(216000)-(pi^2)/(2160)
octahedron tetrahedron picking(19297pi^2)/(3843840)-(6619)/(184320)
sphere tetrahedron picking(4pi)/(105)
tetrahedron tetrahedron picking(13)/(720)-(pi^2)/(15015)

The octahedron tetrahedron picking value is for a regular octahedron of unit volume (Beck 2026).

See also

Mean Line Segment Length, Mean Triangle Area, Tetrahedron Picking

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References

Beck, D. "On Random Simplex Picking Beyond the Blaschke Problem." Methodol. Comput. Appl. Probab. 28, 53, 2026. https://doi.org/10.1007/s11009-026-10277-8. https://arxiv.org/abs/2412.07952.

Referenced on Wolfram|Alpha

Mean Tetrahedron Volume

Cite this as:

Weisstein, Eric W. "Mean Tetrahedron Volume." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MeanTetrahedronVolume.html

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