Microsoft and manhole covers

Many years ago, when I was visiting in the Microsoft Press offices, a discussion of strange employment questions came up. I had heard a story that Microsoft often asked interviewees if they knew why manhole covers were always round. It sounds silly, but there is important logic to round manhole covers - they're round so they can't slip through the hole in the street and cause real problems when they fall into the manhole.

Supposedly they asked this question to see how well people could think outside the box, and of course, programmers who could think outside the box were more desirable to Microsoft. I've never been asked this question about manhole covers by Microsoft or anyone else, but for several years now I've had my own unique answer ready to go. (Microsoft Press published quite a few of my books though, so I'm sure not complaining.)

You see, the standard answer is wrong! There are an infinity of manhole cover shapes that won't fall down the hole. Let me explain.

A circle has a constant diameter, no matter which way you turn it. A square lid, on the other hand, can be rotated such that its width is less than its corner-to-corner distance, allowing it to slip through the square hole in the street.  The theory is that any planar shape other than a circle will have some distance across it that is less that some other distance across it, so it can be turned to slip through its own hole.  But consider the shield shape shown here.

This shape is formed by striking three intersecting arcs with their centers at the corners of an equilateral triangle. The distance from any corner to any point on the far edge is constant, just like the diameters of a circle are all constant.  There is no way to turn this figure such that it can slip through a slot based on its shape, again just like with a circle.

What's more, instead of three equally spaced points to define the corners, we can use 5. Or 7. Or any odd number of points.  Here's what the 5 pointed shield figure looks like.

There are an infinity of figures we can form this way, although they quickly approach the shape of a circle as the number of points increases.