## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## March 24, 2007

### Covering Up

Have you ever wondered why the cover of a manhole is nearly always round? Why couldn't it be oval or square?

The usual answer is that a circular lid, unlike a square or an oval cover, won't fall through the opening. There's no way to position a round lid so that it would slip through a slightly smaller hole of the same shape. That's because the circle has a constant width—it's the same width all the way around.

In contrast, an oval is longer than it is wide. You can always find a way to slip an oval lid through a hole of the same shape. That's also true of a square or a six-sided, or hexagonal, cover.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. Another possibility is the Reuleaux triangle, named after engineer Franz Reuleaux, who was a teacher in Berlin, Germany, more than a hundred years ago. An example of a Reuleaux triangle can be found in your medicine cabinet. If you turn a bottle of NyQuil or Pepto-Bismol upside down, the shape you see looks like a Reuleaux triangle.

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Place the pointed end of a pair of compasses at one corner of the triangle and stretch the arms until the pencil reaches another corner. Then draw an arc between two corners of the triangle. Draw two more arcs centered on the triangle's other corners.

This "curved triangle," as Reuleaux called it, has a constant width—just like a circle. It would certainly work as a manhole cover.

In fact, you can make a manhole cover out of any shape with an odd number of sides. Beginning with a five-sided shape called a pentagon, for example, you can construct a rounded pentagonal shape that has a constant width.

Imagine walking down the street and finding differently shaped manhole covers on every block!

Muse, July/August 1999, p. 36.