I'm trying to find the volume of some arbitrary concrete tetrapod (the things they use to break waves on a beach) and I've broken the problem into a few parts. There's the 4 partial cones and the tetrahedron in the middle (both easy to find), but the thing I'm struggling with is the volume of the overlapping "corners" between the cones.
I've managed to find similar questions about cutting a cone with a vertical plane and finding the volume of a cone using parabolic slices, but in this case I'm dealing with elliptical sections. My current thinking is to maybe use triple integrals in cylindrical coordinates or something, but I have no clue how to set that up.
To give some specifics: each partial cone in the model I've got has an equal diameter and height of 1 with a side slope of 4 (so ~14 deg, but this angle was sort of arbitrary and can be changed a bit if that would make calculations easier). The angle between cones is acos(1/3) (~109.5 deg).
My 3D modelling program gives a total volume of 1.797 units^3 and the volume of a single corner as 0.009 units^3 (this is as accurate as I can get, I think), but ideally I'm looking to make some sort of formula to find volume for a given base diameter.