How many sides does a circle have?

The answer depends on the definition of the word "side." I think this is a terrible question (edit: to put on a quiz) and is the kind of thing that will make children hate math. "Side" is a term that should really be reserved for polygons.

My third-grade son came home a few weeks ago with similar homework questions:

How many faces, edges and vertices do the following have?

• cube
• cylinder
• cone
• sphere

Like most mathematicians, my first reaction was that for the latter objects the question would need a precise definition of face, edge and vertex, and isn't really sensible without such definitions.

But after talking about the problem with numerous people, conducting a kind of social/mathematical experiment, I observed something intriguing. What I observed was that none of my non-mathematical friends and acquaintances had any problem with using an intuitive geometric concept here, and they all agreed completely that the answers should be

• cube: 6 faces, 12 edges, 8 vertices
• cylinder: 3 faces, 2 edges, 0 vertices
• cone: 2 faces, 1 edge, 1 vertex
• sphere: 1 face, 0 edges, 0 vertices

Indeed, these were also the answers desired by my son's teacher (who is a truly outstanding teacher). Meanwhile, all of my mathematical colleagues hemmed and hawed about how we can't really answer, and what does "face" mean in this context anyway, and so on; most of them wanted ultimately to say that a sphere has infinitely many faces and infinitely many vertices and so on. For the homework, my son wrote an explanation giving the answers above, but also explaining that there was a sense in which some of the answers were infinite, depending on what was meant.

At a party this past weekend full of mathematicians and philosophers, it was a fun game to first ask a mathematician the question, who invariably made various objections and refusals and and said it made no sense and so on, and then the non-mathematical spouse would forthrightly give a completely clear account. There were many friendly disputes about it that evening.

So it seems, evidently, that our extensive mathematical training has interfered with our ability to grasp easily what children and non-mathematicians find to be a clear and distinct geometrical concept.

(My actual view, however, is that it is our training that has taught us that the concepts are not so clear and distinct, as witnessed by numerous borderline and counterexample cases in the historical struggle to find the right definitions for the $V-E+F$ and other theorems.)

I know I'm late to the party, but I'm surprised noone has mentioned this. In convexity theory, there is a notion called an extreme point that generalizes the notion of vertex (or corner) of a polygon. For this definition every point on a circle is an extreme point so it makes sense to say it has infinitely (uncountably!) many corners. Though the notion of side is not as good. If the definition is line segment joining two vertices then the answer would be 0 for the circle.