Volume of a tetraedron in function of the areas

[Bruno Tolentino]

Given any tetraedron, I want to calculate the volume V in function of the areas of the surfaces of the solid. I found this pdf that explain this:

http://daylateanddollarshort.com/mathdocs/Heron-like-Results-for-Tetrahedral-Volume.pdf

But, o pdf says that beyond of the 4 faces (X, Y, Z, W) is necessary more 3 pseudo-faces (H, J, K). But, is it correct? Is really necessary 7 areas for compute the volume? With just 4 is not possible?

 

[Simon Bridge]

Have you tried working it out for the areas of 4 faces alone?

The trouble is that the area of a triangle does not determine it's shape - so you can construct many differently volumed tetrahedra out of four triangles knowing only their areas (but not their shapes).