how to calculate the volume of tetrapod?

I want to calculate the volume of a tetrapod , but I couldn't find any idea. here is a simulation of the volume :

• Is there a standard definition of a tetrapod? Each picture I'm finding is somewhat different than all the others. – Carser Aug 20 '16 at 12:10
• A possible definition: the union of four conical frustums of some base radius, top radius and height, the bases arranged in a tetrahedron with each base some distance away from the centre. (Oh, and fill in the space in the middle.) – Parcly Taxel Aug 20 '16 at 12:18
• @mounir ben salem By the Solidworks you used, give to reach the volume. But congratulations on wanting to go beyond the easy ... – LCarvalho Jul 17 '17 at 14:23

If $r$ is the sphere radius, then tetrahedron edges have have length $2\sqrt2 r$, the circles forming the frustums bases have radius $a=(\sqrt6/3) r$, while the height of each cap is $h=r(\sqrt3-1)/\sqrt3$.
One can then apply standard formulas to get the volume of the sphere without caps: $$V={4\over3}\pi r^3\left({8\over 3\sqrt3}-1\right).$$ To this the volume of four frustums must be added, but that depends on their height and upper base radius.