tetrahedron volume given rectangular parallelepiped

Question

Let $A$ be a rectangular parallelepiped with edges of lengths $15, 20, 30$. Let $B$ be a tetrahedron on four non-adjacent vertices of $A$ (i.e no two vertices of $B$ share a common edge of $A$). Compute the volume of $B$.

This site gives a way to calculate but there's gotta be a closed form elegant formula for this.

Answer 1

Let the rectangular parallelepiped measurements be $a,\,b,\,c$ then the mixed product of the three vectors is $6$ times the volume of the desired tetrahedron $$V=\frac16\operatorname{abs} \left| \begin{array}{ccc} a&b&0\\ a&0&c\\ 0&b&c \end{array} \right|=\frac13 abc$$ Thus the desired volume is $\frac13\cdot 15\cdot 20\cdot 30=3000$.