# 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. Apr 8, 2011 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. Apr 8, 2011 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. Apr 8, 2011 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? Apr 8, 2011 at 20:28
• @Douglas Zare: C? Apr 9, 2011 at 18:24

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. Apr 8, 2011 at 19:35
• Or the more common interpretation of the question, "2 sides, inside and outside" Apr 8, 2011 at 19:39
• 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. May 20, 2011 at 10:09
• @Asaf: I really do not think there is a need to be so incredibly precise about what I mean by "quiz." May 20, 2011 at 10:55
• @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. May 20, 2011 at 11:27

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. May 20, 2011 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. May 20, 2011 at 13:45
• I have long thought you must be great to have around at parties :) May 20, 2011 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. May 20, 2011 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, 2011 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. Jul 30, 2013 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... Oct 11, 2011 at 5:44
• @Douglas: 1.Define "line"? 2. Define "Gluing" Aug 26, 2012 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. Nov 29, 2012 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? Oct 13, 2018 at 9:21
• 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). May 4, 2020 at 5:05

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)

• -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. Sep 29, 2015 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?

• Why could it not be 3 sides that happen to coincide? May 20, 2011 at 13:57
• Or four sides, three of which have size 0. May 20, 2011 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. May 20, 2011 at 14:16

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, 2015 at 8:24
• Yes. I think you can proof this by using a homeomorphism from the triangle to the circle. Aug 9, 2015 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, 2015 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".

The question assume Euclidean geometry, What we call $$d_2$$.

In Euclidean geometry, the distance between two points is given by:

$$\sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$$

But this formula can be generalized for more distances:

$$d_p=\sqrt[p]{\vert x_a-x_b\vert^p+\vert y_a-y_b \vert^p}$$

Then we have two special cases:

$$d_1=\vert x_a-x_b\vert+\vert y_a-y_b \vert$$

Which is the Manhattan distance.

$$d_\infty=\sqrt[\infty]{\vert x_a-x_b\vert^\infty+\vert y_a-y_b \vert^\infty}=\max(\vert x_a-x_b\vert,\vert y_a-y_b\vert)$$

Which is the max distance.

Manhattan distance will give you a circle balancing on one apparent vertex. Max distance will give you a circle lying on one of it's four apparent sides.

$$p$$ can take any real value in $$[1,\infty)$$ and the shape would still be a circle, though visually, it would morph between two "squares".

The reality is that you've to define "side" properly. It's where you've to abandon the visuals as they are utterly misleading. And intuition is also really good a misleading.

This is a cautionary tale that even the simplest thing is just a nightmare to work with.

Also, I didn't mention the range $$]0,1)$$ because we obtain strange figures.

The formula for a circle with its center at the origin and of radius one is given by:

$$\sqrt[p]{\vert x\vert^p+\vert y\vert^p}=1$$

You can toy with the circle here: https://www.geogebra.org/calculator/hdguujtr

If you find that "weird", there is even a metric defined as:

$$\cases{1\text{ if both points are different}\\0\text{ if both point are the same}}$$

It's a good exercise to try to think about how would a circle of radius one look like in that metric. (it's rewarding a mind bending)

All three can be correct with explanation--guessing is not correct (also bad teacher regardless--as a former math teacher, I've seen it and it's unfortunate).

infinite (in my opinion the most correct): we can approximate a circle by an $$n$$-regular-polygon--as the the number of sides goes to infinity, the polygon approaches a true circle.

1: an $$n$$-polygon has $$n$$ sides--a circle has a single "straight" edge--a boundary...just as all polygons have a boundary--suggesting that polygons in fact have a single side (inside or out).

0: ahhhh zero--the answer to every counting question if you frame it properly...Just as above, a circle is not a polygon--I can't define its sides, therefore it has exactly "no sides", zero.

and undefined: basically the more honest answer to "zero". Edges are only defined for polygons--since a circle is not a polygon, the question is undefined.

Personally the only correct answer (to me) is infinity as it's the only sane answer. "0" and "undefined" simply skirt the question and basically say that shapes are shapes and are not polygons even though we absolutely know, empirically, that absolutely any shape can be approximated by a polygon with enough and small enough sides. "1" as the answer makes almost no sense to me because I don't know how you define that a polygon has several sides (edges) yet a circle or curved surface has only one (what are you saying is an edge/side then?)

In my humble opinion, all answers from zero up to infinity (as long as it's a natural number) are acceptable: if you define a triangle having three sides (straight segments) and a rectangle having four sides (straight segments), then the amount of sides is zero indeed.

If you want to know the amount of lines (straight or curved) you need in order to draw a circle, then the answer might be one, but this also means that a triangle or a rectangle also might have just one side.

The infinity answer is correct, because a circle is the infinite limit of a regular polygon, consisting of $$n$$ sides (straight segments).

... but one thing is not mentioned here: a triangle is said consisting of three sides (straight segments), but this is not true at all: triangles consist of at least three sides, hereby an example of a triangle with three sides: One with four sides: One with five sides: => that's how difficult mathematics become if you don't use strict definitions!

A Line Segment contain infinite points and Sinc you are talking about a side then we can treat the side as Line Segment.

If we take a regular n-sided polygon where n tends to infinity then sides or Line Segment will approaches to point but not exactly point and in that way the regular polygon approaches to circle but not exactly becomes the circle.

Note: In geometry , the line, point are undefined terms so we can't claim about the Number of sides of Circle.

You should note that Point is dimension less means it has no length, breadth ,height and it is infinitely small quantities and these infinite points makes line so if we choose any two points on line say A and B and claim that distance between A and B is infinitely small then can we say that single point will lies Between them? DEFINATELY NOT, because we can put more infinite small points between A and B.

• -1 I find the "bold" and all-caps to be quite rude. Sep 23, 2022 at 8:26