There is a straight-away answer but I don't know how in-depth it is. It's called Tartaglia's formula and can be found here Niccolò Fontana Tartaglia - Wikipedia, the free encyclopedia

So, let the vertices of the tetrahedron pyramid be 1,2,3 and 4. If [LaTeX ERROR: Convert failed] distance between vertices i and j, then these are just [LaTeX ERROR: Convert failed] .

Then the volume is:

I don't know how one can derive it.