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The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

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Abstract.

 We calculate E[V 4(C)], the expected volume of a tetrahedron whose vertices are chosen randomly (i.e. independently and uniformly) in the interior of C, a cube of unit volume. We find

$$E[V_4 {\rm (cube)}]={3977\over 216000} - {\pi^2 \over 2160}=0.01384277574\ldots$$

The result is in convincing agreement with a simulation of 3000 · 106 trials.

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  1. Parma, Italy, , , , , , IT

    Alessandro Zinani

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  1. Alessandro Zinani
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Received February 12, 2002; in revised form August 13, 2002 Published online February 28, 2003

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Zinani, A. The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube. Monatsh. Math. 139, 341–348 (2003). https://doi.org/10.1007/s00605-002-0531-y

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  • DOI: https://doi.org/10.1007/s00605-002-0531-y

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