how to calculate the volume of tetrapod?

Question

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

Answer 1

A simple possible tetrapod is formed by an inner sphere and four equal cone frustums built on the faces of a tetrahedron (see picture below, where cones are shown instead of frustums). The sphere is tangent to tetrahedron edges and to obtain the volume of tetrapod one must add the volume of the sphere with the volume of the frustums, but subtract the volume of the four spherical caps (pink colored in the picture).

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.