Cayley–Menger determinant

Summary

The Cayley–Menger determinant of $n + 1$ points is a determinant built only from the squares of their pairwise distances. Up to a constant factor, it equals the square of the $n$-dimensional volume of the simplex with these points as vertices. It lets one compute volumes, and test whether given distances fit points of Euclidean space, without coordinates. Heron's formula for the area of a triangle is its case $n = 2$.