Take the equation of a sphere with the center and the radius r:

You need at least 4 vertices of the solid.

Plug in the coordinates of the vertices into the general equation.

After subtracting one equation from the other three equations (the square values will vanish) you have a system of three linear simultaneous equations in .

Afterwards you can calculate the length of r.

That depends on the kind of solid and it would be definitely a very special case.... orbasicallyany 3D object? ...