The napkin ring problem

Keith Devlin writes about The Napkin Ring Problem in his column this month. The problem is the following: consider a sphere with a cylindrical hole drilled out from the center. What is the volume of the resulting solid? At first one thinks that this should depend on the radius of the sphere and the radius of the hole, but it turns out to depend only on the height of the resulting solid; Devlin gives the computation. As it turns out, the volume is 4πh³/3, where h is the half-height of the resulting solid; this is the volume of a sphere of radius h. In fact, the corresponding cross-sectional areas of a "napkin ring" of half-height h and a sphere of radius h are the same, as one can see without calculus; by Cavalieri's principle the volumes are equal.

This is a problem that is usually assigned in our calculus courses when we get to solids of revolution. The way that it's phrased in Stewart's calculus text (which I don't have at hand right now) is in two parts. The first part refers to two such napkin rings -- call them A and B -- and says that napkin ring A comes from a larger-radius sphere than B but also has a larger-radius hole; the student is asked to guess whether A or B has greater volume.

At this point the students invariably ask "what if my guess is wrong?" I laugh and tell them that it doesn't matter; the point is that they should try to think about it for a moment before they plunge into the calculations.

The second part, if I remember correctly (my officemate has the book, so I'll check this tomorrow) asks the students to express the volume as a function of the two radii and the height; often, even if they manage to get the correct answer (which is a bit tricky) they do not seem surprised by this fact, which I was when I first saw it. Then again, it is often difficult to judge whether students are surprised by some statement made in class or on the homework; it's not the sort of thing your average college student is going to let show! (Penn students taking the calculus courses are mostly pre-professional, either in the engineering or business schools; I wonder if the future mathematicians or even the future "pure" scientists would react differently.)

(Incidentally, it is always amusing to look at reviews of textbooks at amazon.com; I often wonder how many of them are written by students who are disgruntled about their class and want somewhere to take it out.)