I may be missing something here, as I was a Literature Major not a Math guy, but wouldn't you need 1 more piece of info regardless of semantics?

If you mean by "a six inch hole through a sphere" a six inch long hole "through" the sphere(all the way), then you need to know the diameter of the hole. We know the diameter of the sphere is 6 inches plus pi * h*h * (r - h / 3) , but the hole could be from microscopic to almost 6 inches plus pi * h*h * (r - h / 3) in diameter; thus, size matters.

If you mean by "a six inch hole through a sphere" a six inch diameter hole through a sphere of unknown size, then you need to know the diameter of the sphere. It can be from a little over 6 inches to some outrageously large finite number; thus, size matters.

The moral of the story is: your girlfriend lied. Size DOES matter.

But seriously, if the clarification above, in fact, means that it's a 6 inch hole THROUGH a sphere, we know only that the sphere must be 6 inches plus pi * h*h * (r - h / 3) in diameter, or one of two things are incorrect: either the hole is NOT 6 inches long, or it's not THROUGH the sphere - it is merely INTO the sphere. But we still don't know the diameter of the hole. IF it is say 5.9" in diameter, then we've made a really cool hat and can figure out the volume. If it's, say, 1" in diameter then we have a giant bead on our hands and can figure out the volume. But without this extra piece of knowledge, the answer set, HAS to be somewhere between > 0 and 4/3 pi times (6 inches plus pi * h*h * (r - h / 3)) cubed.