Volume of a convex hull of 4 points in 3 dimension

Question

I have a problem to calculate the volume of the convex hull of 4 points $\in \mathbb{R}^3$. I've read here https://en.wikipedia.org/wiki/Polyhedron#Volume a formula that is resulted from the divergence theorem.

First one i can't see how it resulted from the divergence theorem.

Second, is there a more simple formula or algorithm to find the volume of the convex hull ?

Answer 1

You are looking for the volume of a tetrahedron. If the $4$ points are $P_i = (x_i, y_i, z_i)$, $i=1,4$, then the volume is $\frac{1}{6}$ of the absolute value of the determinant $$ \left | \matrix{ 1 & x_1 & y_1 & z_1\\1 & x_2 & y_2 & z_2 \\1 & x_3 & y_3 & z_3 \\ 1 & x_4 & y_4 & z_4 } \right | $$