# 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.

• Ridiculous teacher, IMO. – Aryabhata Apr 8 '11 at 19:30
• It sounds very likely that the teacher did not make clear what they considered a "side" to be and whether or not the term is applicable to circles. – Ryan Budney Apr 8 '11 at 19:34
• The question is incomplete as stated. Saying that a triangle has 3 sides and a rectangle has 4 sides is not a good definition of "sides." This is quite a ridiculous question for 2nd grade students. The question can only confuse, and has no definite answer based on this definition. – fdart17 Apr 8 '11 at 19:42
• Clearly the teacher thinks that $lim_{n \to +\infty}n = 1$. Does the teacher have an account somewhere that I can downvote? – knucklebumpler Apr 8 '11 at 20:28
• @Douglas Zare: C? – Fixee Apr 9 '11 at 18:24

## 9 Answers

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.

• I don't think the question is terrible in itself, but asking it without realizing that there are arguments in favour of $0$, $1$ and $\infty$ and marking $\infty$ as wrong is catastrophic. – joriki Apr 8 '11 at 19:35
• Or the more common interpretation of the question, "2 sides, inside and outside" – picakhu Apr 8 '11 at 19:39
• @Donotalo: again, it really depends on the definitions. In my opinion "side" should be restricted to polygons. One should define a polygon as a simple closed piecewise-linear curve in the plane with finitely many linear pieces and the number of sides of a polygon as the number of linear pieces. I don't really see the point of extending this definition beyond the piecewise-linear case. – Qiaochu Yuan Apr 9 '11 at 4:35
• I don't think that this is a terrible question. The terrible thing is to pretend there is a unique answer. Instead, this kind of thing can be a motivation to explain the (non-unique) nature of generalizations and the nature of precise definitions. – Phira May 20 '11 at 10:09
• @Asaf: I really do not think there is a need to be so incredibly precise about what I mean by "quiz." – Qiaochu Yuan May 20 '11 at 10:55

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.)

• "... our extensive mathematical training has interfered with our ability to grasp easily what children and non-mathematicians find to be a clear... concept." - I believe this largely applies to such "find the (integer) pattern" problems as can be found on IQ tests and the like. The mathematicians claim infinite solutions; the non-mathematicians actually fill in an answer! Thanks for your answer, by the way. – The Chaz 2.0 May 20 '11 at 13:37
• @The Chaz: that's a bad example. Your ability to answer such questions is not necessarily related to any objective measure of intelligence: it correlates with being familiar with certain examples of such questions and more generally with being raised in a culture where such questions exist. When mathematicians react negatively to the use of such questions, they are in part reacting to this arbitrariness (at least I am). Being able to answer such questions indicates that you are good at anticipating what kind of answers the makers of the test want, nothing more and nothing less. – Qiaochu Yuan May 20 '11 at 13:45
• I have long thought you must be great to have around at parties :) – Mariano Suárez-Álvarez May 20 '11 at 13:48
• I think this answer could be a great upstep to an interesting discussion on the balance between intuition and rigor, and how the latter may sometimes hinder mathematical advancement. However, I must agree with Qiaochu Yuan that in the particular case of a teacher asking such a question it seems more appropriate to begin a discussion about the lack of mathematical skill and understanding displayed by teachers and how that hinders mathematical advancement. – Myself May 20 '11 at 13:54
• Mariano, thanks for the vote of confidence! (And next time you are in New York, please let me know.) – JDH May 20 '11 at 14:13

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.

• I had my downvotes at the ready, expecting the usual tripe spouted by someone digging up an ancient question - but great answer. – user1729 Jul 30 '13 at 19:49

This is in reference to Douglas Stones' answer, but images can't be imbedded in comments. Limits of sides can have a straight angle, such as these octogons converging to a square. A straight line could be any number of sides with straight angles between them.

For those who are thinking that the answer is $$\lim \limits_{n \rightarrow \infty} n = \infty$$, via:

• An $$n$$-gon has $$n$$ sides;
• A circle is a limit of a $$n$$-gon as $$n \rightarrow \infty$$;
• Therefore a circle has $$\lim \limits_{n \rightarrow \infty} n = \infty$$ sides;

I'd like to mention: it's not so straightforward. If taking limits in this way were legitimate then we can show that e.g. a square has an infinite number of sides.

Consider a staircase with $$n$$ steps, and each step has height $$1/n$$ and width $$1/n$$. It consists of $$2n$$ line segments. As $$n \rightarrow \infty$$, the staircase converges to a single line segment (i.e. the limit agrees point-for-point with a single line segment).

If we glue four of these staircases together, and take their limit, we obtain a square, which would have $$\lim \limits_{n \rightarrow \infty} 4 \times 2n = \infty$$ sides.

• Hehehe... – J. M. is a poor mathematician Oct 11 '11 at 5:44
• @Douglas: 1.Define "line"? 2. Define "Gluing" – Zeta.Investigator Aug 26 '12 at 18:44
• @Douglas: I think the reasoning was not like that, but on the line of "a triangle has three tangents, a square has four, and a circle has infinitely many. – Martin Argerami Nov 29 '12 at 10:20
• @DouglasS.Stones If taking limits in this way were legitimate then we can show that e.g. a square has an infinite number of sides. is a good example of similar flaw, but, not clear where the flaw or what the flaw is. Can you please explain the logical flaw then? Why we can't take limit here? – tarit goswami Oct 13 '18 at 9:21

Personally I use to think a circle had infinite sides as well; however, why could it not be one side with a $$360^\circ$$ curve?

• Why could it not be 3 sides that happen to coincide? – Myself May 20 '11 at 13:57
• Or four sides, three of which have size 0. – Myself May 20 '11 at 13:57
• Because, as I said, the answer depends on the definition of the word "side." The most restrictive definition is that such a thing has to be straight, and in that definition there are no sides. – Qiaochu Yuan May 20 '11 at 14:16

Both answers 1 and $\infty$ are intuitively correct.

To the answer "$\infty$": Imagine that you start with circle. Now you can try approximate the circle by a centered (at middle of circle) hexagon. The next step is to double the number of corners to a regular dodecagon and so on. What you see geometrically is that the $n$-th regular polygon by this construction will approximate the circle better than the $(n-1)$-th one. You can look now at the number of sides during this approximation by doubling the number of corners: $6\to12\to24\to48\to96...\to6\cdot2^n=3\cdot2^{n+1}$. Taking $n\to\infty$ you see that you get $\infty$ sides. (but their length goes to zero...)

To the answer "1": On the other hand it is not intuitive to call it a "side" while its length$\to0$, which is the state in a circle (remember the definition of a circle as a set of points). But what you get is a curved line (the circle itself), which one could interpret as a "side" because it separates the inner region from its environment. And this is one line. This could be the reason for the answer "circle has one side".

However: "$\infty$ or 1?" is a question which causes from the question of the definition of the word "side". (and as one can see "side" makes only really sense for polygons)

• -1 for the "answer could be $\infty$" part. That a figure can be approximated to arbitrary precision by polygons with sufficiently high number of sides is not a good definition of having an infinite number of sides. See the answers by Douglas Stones and robjohn. – epimorphic Sep 29 '15 at 20:35

A circle has indeed $0$ straight sides.

I think the answer to this question relies heavily on the CW structure imposed on $S^1$. I can realise $S^1$ with an arbitrary number of $1$-cells.

• One can realize a triangle with an arbitrary number of $1$-cells. – anon Aug 7 '15 at 8:24
• Yes. I think you can proof this by using a homeomorphism from the triangle to the circle. – ThorbenK Aug 9 '15 at 8:30
• We would never use "number of $1$-cells in a CW structure imposed on the figure" to count the sides of a polygon, so why would we use it to count sides of a circle? – anon Aug 9 '15 at 10:43

## protected by AryabhataMar 23 '12 at 21:41

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