# Tartaglia's Formula

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## Theorem

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

For all $i$ and $j$, let the distance between $\mathbf d_i$ and $\mathbf d_j$ be denoted $d_{ij}$.

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}$

## Proof

A proof of Tartaglia's Formula will be found in a proof of the Value of Cayley-Menger Determinant as a tetrahedron is a $3$-simplex.

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## Also known as

**Tartaglia's Formula** is also known as **Piero della Francesca's Tetrahedron Formula**, for Piero della Francesca.

## Also see

## Source of Name

This entry was named for Niccolò Fontana Tartaglia.

## Sources

- 2009: Karl Wirth and André S. Dreiding:
*Edge lengths determining tetrahedrons*(*Elem. Math.***Vol. 64**: pp. 160 – 170): Theorem $3.1$