Content of Cayley-Menger Determinant

Definition
A determinant that gives the volume of a simplex in $j$ dimensions.

Theorem
If $S$ is a $j$-simplex in $\R^n$ with vertices $v_1,...,v_{j+1}$ and $B=(\beta_{ij})$ denotes the $(j+1)×(j+1)$ matrix given by


 * $\beta_{ij}=|v_{i}-v_{j}|_2^2$,

then the content $V_j$ is given by

where $C$ is the $(j+2)×(j+2)$ matrix obtained from $B$ by bordering $B$ with a top row $(0,1,...,1)$ and a left column $(0,1,...,1)$.

Also see

 * Tartaglia's Formula
 * Heron's Formula