Pick four points at random on the surface of a unit Sphere. Find the distribution of possible volumes of
(nonregular) Tetrahedra. Without loss of generality, the first point can be chosen as (1, 0, 0).
Designate the other points , , and . Then the distances from the first Vertex are

(1) | |||

(2) | |||

(3) |

The average volume is then

(4) |

(5) |

(6) |

(7) | |||

(8) | |||

(9) |

**References**

Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' *Ill. J. Math.* **30**, 653-659, 1986.

© 1996-9

1999-05-26