Can a Generalized Prismatoid Volume be Solved by Axiomatic Means?

[fcsc]

I am trying to solve a volume problem, it is very simple and can be easly solved if the space is parametrized and then integrated.

However, it should also have to be possible to solve the problem by more axiomatic means.

The first version of the problem is: assume that you have a triangle ABC, now you construct a volume on top of this triangle. This volume is build by setting a plane on top of point A at a height of h_A, that goes through a point on top of B and C at heights of h_B and h_C.

Then I want to generalize this problem to have any shape on its basis and assume that the heights will always allow to have a unique plane to define the roof of the volume.

My response, after doing some syntetic geometry on the triangle problem, is:
Volume = AreaOfBasis x ( h_A + h_B + h_C ) / 3
It does make kind of sense to obtain a prism with height being the average of heights.

However, when I add a fourth point (and its height allow us to define a unique plane as a roof) I am not able to deduce the formula.

Any help or hint on how to do it?

 

[Eynstone]

If D is the fourth point, triangulate the quadrilateral ABCD. It might simplify the problem if you express hD in terms of hA,hB,hC or all heights in terms of other parameters.