The Cayley-Menger determinant is a determinant that gives the volume of a simplex in j dimensions. If S is a j-simplex in R^n with vertices v_1, ..., v_(j + 1) and B = (β_(i k)) denotes the (j + 1)×(j + 1) matrix given by β_(i k) = left bracketing bar v_i - v_k right bracketing bar _2^2, then the content V_j is given by V_j^2(S) = (-1)^(j + 1)/(2^j (j!)^2) det(B^^), where B^^ is the (j + 2)×(j + 2) matrix obtained from B by bordering B with a top row (0, 1, ..., 1) and a left column (0, 1, ..., 1)^T. Here, the vector L_2-norms left bracketing bar v_i - v_k right bracketing bar _2 are the edge lengths and the determinant in (-1) is the Cayley-Menger determinant (Sommerville 1958, Gritzmann and Klee 1994).

Heron's formula | quadrilateral | tetrahedron