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.

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10 Answers10


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.

Qiaochu Yuan
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    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 08 '11 at 19:35
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    Or the more common interpretation of the question, "2 sides, inside and outside" – picakhu Apr 08 '11 at 19:39
  • i'm a bit confused. why a circle doesn't have infinitely many sides? it has infinitely many tangents, right? – Donotalo Apr 09 '11 at 02:47
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    @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 09 '11 at 04:35
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    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
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    @user9325: let me clarify. This is a terrible question to put on a _quiz._ (I really thought this was clear from context, but apparently I was mistaken.) – Qiaochu Yuan May 20 '11 at 10:30
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    @Qiaochu: Sorry, you are right, of course. – Phira May 20 '11 at 10:34
  • @Qiaochu: If you're editing, then you should say that it is actually a terrible question for a "closed form" quiz for children. I think that putting that as an open question in a quiz for math undergrads can produce interesting results. – Asaf Karagila May 20 '11 at 10:40
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    @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
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    @Qiaochu: I was actually particular about the audience, not about the definition. Because it doesn't make much sense to put this in a second grade quiz, but might make sense in a philosophy/math undergrad quiz. – Asaf Karagila May 20 '11 at 11:25
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    @Asaf: I disagree. This would be much more productive as a class discussion. It's too much of a trick question to reasonably ask in a setting where the students will be graded on their answers. – Qiaochu Yuan May 20 '11 at 11:27
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    @Qiaochu: This is why I said an open question, I would much prefer to see the students defend their own arguments in a sound and consistent way. – Asaf Karagila May 20 '11 at 11:30
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    Value of $\pi $ Can be found by considering a circle as a nearly infinite sided polygon. – N.S.JOHN Apr 02 '16 at 15:17
  • NS John is right. We can be found Pi number if the number of faces are infinite. Also the more sides you add to a polygon the more it resembles to a circle. Sure you can set this a different case but if you go deeper into analysis, it only makes sense with the rest of mathematics if you treat it as a polygon with infinite number of faces. – Alfonso Fernandez-Ocampo Apr 28 '19 at 11:16

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.)

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    Okay, so these are "manifolds with (manifolds with boundary) as boundaries," faces describe the manifold parts, edges describe the manifold parts of the boundaries, and vertices describe the boundaries of the boundaries. That's fine as far as that goes, but not having formal definitions in this situation is just going to make it harder for you to distinguish between the different types of faces and edges implied by those counts, and consequently you'll have a harder time discovering Euler's formula. What we lose in our ability to answer trick questions we gain in _Euler's formula._ – Qiaochu Yuan May 20 '11 at 13:35
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    "... 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
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    @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
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    I have long thought you must be great to have around at parties :) – Mariano Suárez-Álvarez May 20 '11 at 13:48
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    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
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    Qiaochu, your manifolds-with-manifolds-with-boundaries-boundaries definition is of course very natural, and most of us would be tempted in that direction. But does it give the right calculation for the cone? After all, third-graders and third-grade teachers will insist that a cone has one edge (the bottom circle) and one vertex (the top point), but this vertex is not on the boundary of that edge. (But I suppose one could equivocate about non-empty edges, etc.) – JDH May 20 '11 at 14:04
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    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
  • @JDH: fair enough. How about this: I think the definition of the tangent space at a point via equivalence classes of smooth curves works in this situation, and then I believe it's true that the naive count of faces, edges, etc. counts connected components of the subspaces of points whose tangent spaces have the relevant dimensions. – Qiaochu Yuan May 20 '11 at 14:22
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    @Myself, I agree with your first point, but not your second, as my son's teacher, as I mentioned, is truly outstanding. The homework questions I mention are not a case of mathematical ineptitude (particularly because she was open to the possibility of more sophisticated answers discussing the issue as here, unlike the teacher of the OP above). – JDH May 20 '11 at 14:27
  • @JDH Of course my own comment on 'teachers' was a reference to the OP's story, where a teacher will (apparently) vigorously defend his or her own version of truth, based on some definitions that were never clearly stated. I was certainly not making a statement about teachers in general. – Myself May 20 '11 at 15:11
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    Your writing is beautiful and wonderful. Thank you so much! – user729424 Jan 15 '20 at 16:01
  • @AndrewOstergaard You are very kind to say so; thank you very much. – JDH Jan 15 '20 at 17:28
  • @QiaochuYuan The sequence extrapolation problem can be seen as the problem of finding continuations of a sequence with low [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity). The ability to find such low-complexity hypotheses or explanations [is a measure of universal intelligence](https://www.sciencedirect.com/science/article/pii/S0004370210001554) (see also [Solomonoff induction](https://en.wikipedia.org/wiki/Solomonoff%27s_theory_of_inductive_inference)). – user76284 Mar 26 '20 at 21:42
  • I have posted a question about this [here](https://math.stackexchange.com/questions/3613078/what-captures-our-intuitive-notion-of-faces-edges-and-vertices), in case anyone is interested. – user76284 Apr 16 '20 at 18:45

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.

Owen Sizemore
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    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.

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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.

J. W. Tanner
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Douglas S. Stones
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    [Hehehe...](http://math.stackexchange.com/questions/12906) – J. M. ain't a mathematician Oct 11 '11 at 05:44
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    @Douglas: 1.Define "line"? 2. Define "Gluing" – Zeta.Investigator Aug 26 '12 at 18:44
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    @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
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    @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 09:21
  • This is an excellent response to those who think that a circle has infinite sides. Those people are assuming that 'side' means 'straight edge', whereas my belief is that side means 'smooth edge' (at least in this context). My daughter (8) had the same question in her schoolwork today, and she put 0, but I told her it should be 1 (and yes, I am a mathematician). – Laurence Renshaw May 04 '20 at 05:05

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)

Mister Theory
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    -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

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?

J. W. Tanner
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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.

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    One can realize a triangle with an arbitrary number of $1$-cells. – anon Aug 07 '15 at 08:24
  • Yes. I think you can proof this by using a homeomorphism from the triangle to the circle. – ThorbenK Aug 09 '15 at 08:30
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    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 09 '15 at 10:43

One way to understand the question and demonstrate applicability is to consider the problem and efficiency of finding the area of the union of 2 overlapping circles versus 2 overlapping rectangles or squares.

Let us now constrain the union area to be constant.

In the case of circles it will always be the same shape, no matter which way the circles are positioned in relation to one another.

In the case of squares, it would be the same shape too.

In case of rectangles, the shape would vary.

We could here argue that the circle and square both have 1 side, because they are defined by a single length (radius or diagonal), or in other words, they have no apparent "orientation".

Ate Somebits
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