I first saw his puzzle over 25 years ago when it concerned a mouse eating his way through a ball of cheese. I remember the date as I put the puzzle to my then future wife on St Valentine's day over a romantic(!) dinner date. Bemused, she allowed me to solve it on my napkin using integral of pi y squared dx.
But leaving that aside. Since you're not told of the diameter of the hole it can make no difference, so take the diameter as 0. You then have a sphere of radius 1cm whose volume will be 4/3 pi r cubed, (4.pi)/3.
As I said to her at the time: QED