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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

 Figure 1

Take a bead 5 inches high, and bore a hole through it from north pole to south pole in such a way that the part which remains is 4 inches high (measured lengthwise through the hole).

Now take a bead 5 feet high, and bore a hole through it in the same fashion, so that the part which remains is also 4 inches high (you'll have to bore a very large hole).

Surprisingly, the volumes of the two "holey" beads are now exactly the same! (In fact, the same volume as a 4-inch bead with no hole!) Actually, it doesn't matter how big the bead is that you start with... you could have started with the Moon, and it would still be true.

Presentation Suggestions:
Draw suggestive pictures!

The Math Behind the Fact:
You can assign the proof as a homework exercise in multivariable calculus. Cavalieri's principle might be helpful.

Su, Francis E., et al. "Holey Beads." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Subjects:    calculus, analysis
Level:    Easy
Fun Fact suggested by:   Michael Moody
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