## Scott’s Conundrum

A friend of mine once wondered why manhole covers are round, and made the mistake of entering a query into a search engine that contained the word “manhole”. Although it returned millions of hits, none of them answered the question (HINT: “manhole” is a search that is not work safe).

So, having recently stumbled on some information, I’ve decided to write this to answer conclusively why manhole covers are round.

The reason manhole covers are round is that if you bevel the edges, you cannot put the cover back down the hole. Thus, the workers below can’t get clobbered by a 50 kg chunk of steel. Any other regular* shape (rectangle, square, pentagon, triangle, dodecagon, whatever…) can be turned in such a way as to fit down the hole because it is thinner in some directions than others. For example, a square manhole cover, 1m on a side could be slipped diagonally down the hole because the widest diameter of the whole is 1.4m.

A circle, however, is the same width in any diameter. That is to say, the “widest” diameter of a circle is the same size no matter how you orient the thing. Consequently, the circle cannot be turned so as to be wedged down its hole (if the edges were beveled slightly). A circle is said to be a shape of “constant width”.

For years, that was adequate for me because I don’t sit around and ponder manholes. However, I have to admit I had wondered if there were any other shapes that might also have this property of “constant width”. Well, it turns out that there are. They are called Reuleaux shapes. The simplest of these is the Reuleaux triangle. Simply described, they are regular polygons (n-sided shapes) where the straight sides are replaced by the arcs of circles centred on the opposite corner with a radius equal to the length of a side.

Any shape so constructed will be of constant width – the widest point in any orientation will be the same. Thus, such a shape could not fit down a hole of the same shape (if the hole is slightly beveled). As a result, there is an infinity of possible manhole cover shapes. The wikipedia link above shows the Reuleaux triangle. Here is a picture of a Reuleaux pentagon:

My drawing skills are so-so, but if you measure it, it should be the same width all the way around.

So there you have it. Manhole covers can be any shape that has constant width. The easiest shape is circular, but there are many other possibilities.

* A regular polygon is a multi-sided, two-dimensional shape where all the sides are straight line segments (not arcs), and the internal angles where the sides meet are all the same. A square is a regular polygon because all the sides are straight, are the same length and all internal angles are 90°. A star is not a regular polygon… the sides can be all the same length and straight, but there are different angles where the lines meet.