# Tartaglia's Formula

## Theorem

Let $T$ be a tetrahedron with vertices $\mathbf d_1, \mathbf d_2, \mathbf d_3$ and $\mathbf d_4$.

Denote the respective distances by $d_{12}$ et cetera.

Then the volume $V_T$ of $T$ satisfies:

$V_T^2 = \dfrac {1} {288} \det \ \begin{vmatrix} 0 & 1 & 1 & 1 & 1\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix}$

## Source of Name

This entry was named for Niccolò Fontana Tartaglia.