Problem 8 (Barber Pole)

Consider a cylindrical barber pole 3 feet long and 1 foot in circumference, like the one

in which a strip of paint in the shape of a helix is painted. If the strip starts and ends on the same "cylindrical element" (a vertical line on the cylinder), what is the length of the strip of paint ?

This problem is very easy if you think about it in just the right way.

Solution
There is a very clever solution to this problem. Think of rolling the pole over on the plane 4 times where the paint strip is still wet so it leaves a trace on the plane. It is clear that the paint traces out the hypoteneuse of a right triangle with its legs 3 feet (height of the pole) and 4 x 1 = 4 feet as shown below.

Hence, from the Pythagorean theorem, the length of the paint strip is

length of paint strip = sqrt (3² + 4² ) = 5 feet

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Last modified on Friday, February 12, 1999