By Math Is Fun, January 15, 2022

I was jogging outside with my son on the street and we saw a manhole. I recalled this famous interview question first popularized by Microsoft in the early days and then adopted by other companies: Why are manhole covers round?

So I asked my son.

“Because it looks nice?”

“It’s more than that. There is actually math magic behind it! The main reason manhole covers are made in round shapes is that the cover cannot fall into the manhole. Let’s say construction workers need to work under a manhole for the piping. They first remove the cover, put the cover aside, then go under to do the work. Somebody comes by and hits the cover by accident. If the cover can fall into the manhole, that will be a very bad accident!

“To explain, let’s look at some other common shapes. For example, can the manhole cover be a square? Note the manhole and the manhole cover need to be the same shape. So the question is: can a square cover fall through a manhole of the same shape?”

Son mimicked the shape for a while and found out it could, “Yes it can fall like this.” He meant the cover can fall through the manhole diagonal.

“That’s correct. What about a triangle? Can it fall through a triangle of the same shape?”

“… yes.”

“Now let’s go back to the round shape. Can a circle fall through a circle of the same size?”

“… I am not sure I can find a way…”

“You are actually right. This is an important property of a circle. The width of a circle is always the same, no matter how you turn the circle around. This width is called the diameter of the circle. Since the manhole itself is the same shape, its diameter is the same as the diameter of the cover. So the cover cannot fall through.

“There are still some other shapes that have constant width. For example, there is a shape called Reuleaux triangle. It looks like this:

“And you can actually find Reuleaux triangle shaped manhole covers!

“The manhole cover shape question is an example of Geometry. **Geometry** is a math branch that studies the properties of space, for example, shapes, distances and sizes. The geometric property of the circle – a constant width – is why manhole covers are round. Geometry is one of the oldest math branches, dating back to ancient Greece.”