The Cayley-Menger determinant is a determinant that gives the volume of a simplex in dimensions. If is a -simplex in with vertices and denotes the matrix given by
(1)
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then the content is given by
(2)
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where is the matrix obtained from by bordering with a top row and a left column . Here, the vector L2-norms are the edge lengths and the determinant in (2) is the Cayley-Menger determinant (Sommerville 1958, Gritzmann and Klee 1994).
The multiplicative inverses of the prefactors for , 1, 2, ... are , 2, , 288, , 460800, ... (OEIS A055546).
For , (2) becomes
(3)
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which gives the area for a plane triangle with side lengths , , and , and is a form of Heron's formula.
For , the content of the 3-simplex (i.e., volume of the general tetrahedron) is given by the determinant
(4)
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where the edge between vertices and has length . Setting the left side equal to 0 (corresponding to a tetrahedron of volume 0) gives a relationship between the distances between vertices of a planar quadrilateral (Uspensky 1948, p. 256).
Buchholz (1992) gives a slightly different (and slightly less symmetrical) form of this equation.