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The Napkin Ring Problem
I was with a group of business people recently when one of them brought up the problem of calculating the volume that remains when a circular cylinder is removed from the center of a sphere. Since the remaining figure resembles a napkin ring, this is sometimes called the Napkin Ring Problem. The surprising fact is that the volume does not depend upon the radius, r, of the sphere, but only on the height of the cylinder.
If the removed cylinder has height 2h and radius a, then the napkin ring has volume 4/3 PI h^3. See the cross-section diagram below.
If you know the standard formulas for the volume of a sphere and the volume of a cylinder, all you need to do is use some elementary integral calculus to compute the volume of the circular cap that falls off each end of the sphere when the cylinder is drilled out, and then you can calculate:
Vol of ring = vol of sphere - vol of cylinder - 2 x vol of end cap.
The entire computation is given here.
In the extreme case where the cylinder has height 2r, where r is the radius of the sphere, the cylinder has zero diameter, of course, so no volume is removed from the sphere, and in this case the volume formula reduces to 4/3 PI r^3, the standard formula for the volume of a sphere.
As the only mathematician present, I was asked to explain how this answer is obtained. We were, however, in a restaurant at dinner, and with the main course about to be served, I was reluctant to start scribbling on napkins (for which rings were not provided, as it happens). Instead, I remarked that this was a well known problem in calculus courses, and that they would have no trouble finding the answer on the Web. Search on "napkin ring problem" or else the search terms "volume + sphere + cylinder + removed" I said.
That was the end of the discussion, but afterwards I went online to see what kind of answer my dinner colleagues would find. A search along the above lines does yield a number of hits. I was relieved to find that most of them are correct, though in many cases non-mathematicians might find the explanations hard to follow. But there were some glaringly false answers. In particular, the solution posted at WikiAnswers is
4/3 PI r^3 - 2 PI r^2h
This answer has two terrible errors: first, assuming the cylinder has the same radius as the sphere and second, forgetting to account for the two end caps. My experience emphasized yet again that anyone who uses the Web to find information should exercise caution. Just because something it stated in a professional looking web page, doesn't mean it is necessarily correct. Students please take note.
Finally, in last month's column I reproduced in its entirety an essay on K-12 mathematics education by a New York based math teacher (with a Ph.D.) called Paul Lockhart. The publication of Lockhart's essay generated a large volume of email for me, and apparently a much larger flood for Paul himself. The emails I received ranged from the passionately negative (one writer started his tirade with the words "the guy borders on being a complete whack-job") to the highly enthusiastic. The latter were by far the majority. There were also a number of emails that, while generally in support of the points Lockhart was making, sought to temper or counter particular aspects of his argument, for various reasons. Many of the points raised were related to the fact that, in addition to mathematics' intrinsic beauty as an intellectual pursuit, it is a darned useful tool that many professionals need to have mastery of, but have little interest in, or time to focus on, the subject itself. This connects to the "American Mathematics in a Flat World" issues I raised in my January and February columns. I'll provide a summary of the feedback I received in a later column.
Devlin's Angle is updated at the beginning
of each month.
Mathematician Keith Devlin (email:
firstname.lastname@example.org) is the
Executive Director of the Center for the
Study of Language and Information at
Stanford University and
The Math Guy on NPR's Weekend Edition.
Devlin's most recent book,
Solving Crimes with Mathematics:
THE NUMBERS BEHIND NUMB3RS,
is the companion book to the hit
television crime series NUMB3RS,
and is co-written with Professor Gary Lorden
of Caltech, the lead mathematics
adviser on the series. It was published
in September by Plume.