This entry contributed by Karen D. Colins

A determinant that gives the volume of a simplex in j dimensions. If S is a j-simplex in with vertices and denotes the matrix given by

(1)

then the content is given by

(2)

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 first few coefficients for j = 0, 1, ... are -1, 2, -16, 288, -9216, 460800, ... (Sloane's A055546).

For j = 2, (2) becomes

(3)

which gives the area for a plane triangle with side lengths a, b, and c, and is a form of Heron's formula.

For j = 3, the content of the 3-simplex (i.e., volume of the general tetrahedron) is given by the determinant

(4)

where the edge between vertices i and j 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.