Here 10 cylinders intersect following the corners of a dodecahedra.

With radius 1 the surface of the intersection is

and its volume is a third of the surface.

We have three different not flat corners. The corner with three edges has a solid angle of 46.09%,
the corner with five edges has a solid angle of 41.21% and
the corner with six edges has a solid angle of 34.15%.