How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this:

If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have?

My first reaction was "0" or "undefined". But my son wrote "$\infty$" which I think is a reasonable answer. However, it was marked wrong with the comment, "the answer is 1".

Is there an accepted correct answer in geometry?

edit: I ran into this teacher recently and mentioned this quiz problem. She said she thought my son had written "8." She didn't know that a sideways "8" means infinity.

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.)