# Tartaglia's Formula

Jump to navigation
Jump to search

## 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.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Source of Name

This entry was named for NiccolĂ˛ Fontana Tartaglia.