Content of Cayley-Menger Determinant

Theorem
Let a $k$-simplex be a $k$ dimensional polytope with $k+1$ vertices.

Let $k$ be an arbitrary positive integer dimension and $\lambda_{ij} = \left\vert{v_i, v_j}\right\vert$, for all $0 \le i < j \le k$.

Then for every $k$-simplex $\Delta = \left({v_0, v_1, \ldots, v_k}\right) \subset \R^k$, we have:

Also see

 * Tartaglia's Formula
 * Heron's Formula