How do I measure the volume encapsulated by a plane and an actual mesh.

[[To...]]

Hello,

I have a STL of a dome shape and want to measure the volume of it relative to a plane fitted to the top surface. I can't do this by just fitting geometry elements to the surface, such as fitting a sphere and using that to calculate it, I have to measure it to the actual mesh itself. 

Below are some diagrams:

Top view of sample, dome in the middle, blue line showing cross section for second diagram:

 

Cross section view:

Cross section view with fitted plane show in green and desired volume in red:

Any help would be greatly appreciated. Let me know if I can clarify anything.