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# How many sides does a circle have?

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...and so were Newton and Leibniz confounded.

Zero, one, two and infinite are all defensible answers. It really depends on how you define side.

posted by bonehead at 7:26 PM on December 20, 2006

Nah. "Infinite number of edges" is a fine way to describe a circle. Just as "infinite number of terms" is a fine way to describe the series: 1/2 + 1/4 + 1/8 + 1/16....which nicely adds to a perfectly complete 1.

posted by vacapinta at 7:58 PM on December 20, 2006 [1 favorite]

I think you're overthinking this: the line of the circle divides the plane it is drawn on into two parts: that of finite area inside the line and that of infite area outside the line.

This definition is cheezy because any topologically simple shape (triangle, square, icosahedron, blobby bean-thing) does the same thing. A three-sided shape, by this definition, would be a doughnut (or any shape within a shape).

posted by bonehead at 8:04 PM on December 20, 2006

This cannot be. A circle is a one dimensional curve in a two dimensional space. A Möbius strip is a two dimensional 'curve', or surface, in a three dimensional space.

This argument is definitely an argument concerning definitions, and not concerning the truth of falsity of a statement. Once you two agree on what you both mean by a 'side', I think your respective points will be trivially true by definition.

posted by philomathoholic at 12:15 AM on December 21, 2006

In math discussions your argument should stand for itself. If you need to add bluster, you're likely trying to prop up an argument thats weak to begin with.

I've found this to be a good guideline in math/science ask.mefi threads.

posted by vacapinta at 11:34 AM on December 21, 2006

2 (front & back)

posted by flug at 6:18 PM on December 21, 2006

posted by escabeche at 10:22 PM on December 21, 2006

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Post

# How many sides does a circle have?

December 20, 2006 6:47 PM Subscribe

How many sides does a circle have? I always assumed the answer was an infinite number of sides. My boyfriend disagrees with me and thinks that the answer is zero. My brother seems to think that it depends on your definition of what a side is. I have searched all over but can't seem to find a definitive answer.

I'd say that the answer is, oddly enough, contained in your own user name. A circle has two sides: inside and outside.

posted by Faint of Butt at 6:49 PM on December 20, 2006

posted by Faint of Butt at 6:49 PM on December 20, 2006

One.

posted by Robot Johnny at 6:49 PM on December 20, 2006

posted by Robot Johnny at 6:49 PM on December 20, 2006

Is the circle a hollow line or is it filled? That affects whether there is an inside or an outside...

Essentially, I agree with with your brother, and then some: It's

posted by Robert Angelo at 6:53 PM on December 20, 2006

Essentially, I agree with with your brother, and then some: It's

*all*a matter of definition.posted by Robert Angelo at 6:53 PM on December 20, 2006

As far as I know, a circle has an infinite amount of points on the line of it. If you're speaking in terms of straight sides, well, zero, but there is an infinite amount of tangents (lines that connect at one point on the given object) you could draw around the circle.

posted by stresstwig at 6:53 PM on December 20, 2006

posted by stresstwig at 6:53 PM on December 20, 2006

See also, via Google: "How many sides does a circle have?", is too ambiguous to have a definite answer.

posted by Robert Angelo at 6:54 PM on December 20, 2006

posted by Robert Angelo at 6:54 PM on December 20, 2006

Depends how you look at it. A circle isn't really a polygon by the strictest sense of the word - a polygon is defined as a closed, convex (no sides cross) shape made entirely of line segments. A regular polygon has all of its angles equal and all of its sides of equal length. So, a circle is the shape you get when you increase the number of sides of a regular polygon to infinity (you're taking a limit of n, the number of sides).

The above applies to Euclidian geometry. Other geometries will give you different answers. I believe defining a circle as one line is also correct in some circumstances.

posted by backseatpilot at 6:55 PM on December 20, 2006

The above applies to Euclidian geometry. Other geometries will give you different answers. I believe defining a circle as one line is also correct in some circumstances.

posted by backseatpilot at 6:55 PM on December 20, 2006

A circle can have as many sides as you want it to have :)

posted by -harlequin- at 6:56 PM on December 20, 2006

posted by -harlequin- at 6:56 PM on December 20, 2006

'the question, "How many sides does a circle have?", is too ambiguous to have a definite answer.'

For the record, this was the first google result for "how many sides does a circle have?" Apologies if it doesn't solve your problem, but it seems like as good an answer as any.

posted by The God Complex at 6:56 PM on December 20, 2006

For the record, this was the first google result for "how many sides does a circle have?" Apologies if it doesn't solve your problem, but it seems like as good an answer as any.

posted by The God Complex at 6:56 PM on December 20, 2006

I think this might be cleared up by replacing the word "side" with the word "edge".

One way to describe a circle is to say that it's the limit of a series of regular polygons as the number of edges goes to infinity. I think that's what you were thinking of.

But that doesn't mean it has an infinite number of edges. Limits never complete. (That's what "infinity" means.)

An "edge" is a line between two corners. In the simplistic case used in that limit, they're adjacent corners.

So how many corners does a circle have? It doesn't have any, and that means it doesn't have edges either. It's got a circumference, but that's not the same thing.

The simplistic polygons described above are made up of a series of edges between adjacent corners, with all the corners being equidistant from some single point, the center.

But the true definition of a circle is "all the points on a plane which are some particular distance from a single point, the center". There's no interpolation, no edges, and the points making up the circle are not corners. And that's what your boyfriend is thinking.

...and you're both wrong. A circle has no edges, but it has two sides: the inside and the outside.

posted by Steven C. Den Beste at 6:59 PM on December 20, 2006

One way to describe a circle is to say that it's the limit of a series of regular polygons as the number of edges goes to infinity. I think that's what you were thinking of.

But that doesn't mean it has an infinite number of edges. Limits never complete. (That's what "infinity" means.)

An "edge" is a line between two corners. In the simplistic case used in that limit, they're adjacent corners.

So how many corners does a circle have? It doesn't have any, and that means it doesn't have edges either. It's got a circumference, but that's not the same thing.

The simplistic polygons described above are made up of a series of edges between adjacent corners, with all the corners being equidistant from some single point, the center.

But the true definition of a circle is "all the points on a plane which are some particular distance from a single point, the center". There's no interpolation, no edges, and the points making up the circle are not corners. And that's what your boyfriend is thinking.

...and you're both wrong. A circle has no edges, but it has two sides: the inside and the outside.

posted by Steven C. Den Beste at 6:59 PM on December 20, 2006

I'm not sure about the "inside/outside" thing. I mean, in that case, why wouldn't it be true that a square has eight sides instead of four?

posted by Land Stander at 7:18 PM on December 20, 2006

posted by Land Stander at 7:18 PM on December 20, 2006

It has four sides: the inside, the outside, the front side, and the backside. If you believe this it would seem to have one edge (see claims six and 14).

posted by oddman at 7:21 PM on December 20, 2006

posted by oddman at 7:21 PM on December 20, 2006

*Limits never complete.*

...and so were Newton and Leibniz confounded.

Zero, one, two and infinite are all defensible answers. It really depends on how you define side.

posted by bonehead at 7:26 PM on December 20, 2006

I don't think two is a defensible answer, unless there's something I'm missing in this thread. Do squares have 8 sides?

posted by Hildago at 7:39 PM on December 20, 2006

posted by Hildago at 7:39 PM on December 20, 2006

two: inside and outside. Cheezy, I know, but depends on the definition.

posted by bonehead at 7:42 PM on December 20, 2006

posted by bonehead at 7:42 PM on December 20, 2006

*But that doesn't mean it has an infinite number of edges. Limits never complete. (That's what "infinity" means.)*

Nah. "Infinite number of edges" is a fine way to describe a circle. Just as "infinite number of terms" is a fine way to describe the series: 1/2 + 1/4 + 1/8 + 1/16....which nicely adds to a perfectly complete 1.

posted by vacapinta at 7:58 PM on December 20, 2006 [1 favorite]

I vote for one side. The inside/outside answer is what I would expect from a brainteaser or trick question.

posted by Joh at 8:00 PM on December 20, 2006

posted by Joh at 8:00 PM on December 20, 2006

*why wouldn't it be true that a square has eight sides instead of four?*

I think you're overthinking this: the line of the circle divides the plane it is drawn on into two parts: that of finite area inside the line and that of infite area outside the line.

This definition is cheezy because any topologically simple shape (triangle, square, icosahedron, blobby bean-thing) does the same thing. A three-sided shape, by this definition, would be a doughnut (or any shape within a shape).

posted by bonehead at 8:04 PM on December 20, 2006

It has one side. That's why you can't be on the other side of it. Sheesh.

posted by bingo at 8:12 PM on December 20, 2006

posted by bingo at 8:12 PM on December 20, 2006

stevis23:

And if it has a tired sci-fi plot and she-mullets, it's Silent Möbius.

But seriously, I'd say an infinite number of sides is a perfectly acceptable lay description of a circle.

posted by Spike at 9:23 PM on December 20, 2006

*If it only has one side it's a Möbius strip*And if it has a tired sci-fi plot and she-mullets, it's Silent Möbius.

But seriously, I'd say an infinite number of sides is a perfectly acceptable lay description of a circle.

posted by Spike at 9:23 PM on December 20, 2006

karlscalculus says:

Back in high school you learned that the area of a circle is πr2. But did you ever wonder how we know this? Archimedes was the first to figure it out.

So how did he do it?

The figure on the right shows a circle with polygons inscribed. As the animation progresses, the frames show polygons with 4, 8, 16, 32, 64, and 128 sides. You can see that when the polygon has more sides, its area is a bigger fraction of the circle's area. And you can see that in the limit

-- i don't know who karl is, but he and archimedes seem to say "infinite, pretty much". if the polygon had a billion sides, it would still technically not be a circle, although you would never be able to tell.

reminds me of "the koch curve", which shows that within a finite area there can be a closed loop of infinite length

posted by white light at 9:41 PM on December 20, 2006

Back in high school you learned that the area of a circle is πr2. But did you ever wonder how we know this? Archimedes was the first to figure it out.

So how did he do it?

The figure on the right shows a circle with polygons inscribed. As the animation progresses, the frames show polygons with 4, 8, 16, 32, 64, and 128 sides. You can see that when the polygon has more sides, its area is a bigger fraction of the circle's area. And you can see that in the limit

**You know this because any point you choose inside the circle is eventually included inside some polygon with sufficient number of sides. Based upon this, a process Archimedes called "exhaustion", he concluded that***as the number of sides goes to infinity, the polygon occupies all of the circle.***To compute the area of the polygon, he divided it into triangles (as is also shown in the figure), one triangle for each of the polygon's sides.***the area of the circle was the limit of the area of the inscribed polygon as the number of sides goes to infinity.*-- i don't know who karl is, but he and archimedes seem to say "infinite, pretty much". if the polygon had a billion sides, it would still technically not be a circle, although you would never be able to tell.

reminds me of "the koch curve", which shows that within a finite area there can be a closed loop of infinite length

posted by white light at 9:41 PM on December 20, 2006

oh! that "karlscalculus" is an external link to a site with diagrams. it may read as if i'm quoting a mefi user with it being at the beginning of my post like that.

posted by white light at 9:43 PM on December 20, 2006

posted by white light at 9:43 PM on December 20, 2006

A circle is a set of points equidistant from the center point along a given plane.

A side is the linear edge of a polygon.

Circles have no linear edges, as they are simply a collection of points, not a collection of lines. An equilateral polygon with a billion sides might look a lot like a circle, but it isn't one.

As such, no sides.

Sorry, but you missed on this one. Better luck next time.

posted by Tacos Are Pretty Great at 9:50 PM on December 20, 2006

A side is the linear edge of a polygon.

Circles have no linear edges, as they are simply a collection of points, not a collection of lines. An equilateral polygon with a billion sides might look a lot like a circle, but it isn't one.

As such, no sides.

Sorry, but you missed on this one. Better luck next time.

posted by Tacos Are Pretty Great at 9:50 PM on December 20, 2006

*If it only has one side it's a Möbius strip*

This cannot be. A circle is a one dimensional curve in a two dimensional space. A Möbius strip is a two dimensional 'curve', or surface, in a three dimensional space.

This argument is definitely an argument concerning definitions, and not concerning the truth of falsity of a statement. Once you two agree on what you both mean by a 'side', I think your respective points will be trivially true by definition.

posted by philomathoholic at 12:15 AM on December 21, 2006

....what about a bagel? doesn't have an edge,and it's a circle.

posted by hortense at 12:33 AM on December 21, 2006

posted by hortense at 12:33 AM on December 21, 2006

A point is a line of infinitely small length, hence a circle is a polygon with an infinite number of sides, each of them infinitely small.

posted by No Mutant Enemy at 1:02 AM on December 21, 2006

posted by No Mutant Enemy at 1:02 AM on December 21, 2006

Actually hortense... a bagel is technically a torus.

And to answer this question, we really need to know what the context is that you're asking about. If it's a brainteaser, then the answer is probably 2 (inside/outside). If "side" is defined as "straight edge" then your answer could be 0 or infinity, depending on how smart you want to be about it (see No Mutant Enemy's response).

posted by antifuse at 3:33 AM on December 21, 2006

And to answer this question, we really need to know what the context is that you're asking about. If it's a brainteaser, then the answer is probably 2 (inside/outside). If "side" is defined as "straight edge" then your answer could be 0 or infinity, depending on how smart you want to be about it (see No Mutant Enemy's response).

posted by antifuse at 3:33 AM on December 21, 2006

In an analogous problem - how many 9s are there in 0.9999.... ? An infinite number, unless I skip straight to what the limiting process of adding all those 9s in the decimal together is, in which case I get 1, and then I've got no 9s in it. It depends how I construct my final answer. Two different ways of expressing the same result.

posted by edd at 5:16 AM on December 21, 2006

posted by edd at 5:16 AM on December 21, 2006

IAAMathematician.

As many people have told you, the question is not very well-formed since you have not defined what you mean by a "side." If you mean by a "side" a line segment of positive length, then of course there are none. But this is not a great definition -- for instance, a triangle, or any other polygon, contains

A better definition, though not the one you might first think of, is: "a side is a line which intersects the circle but does not contain any points interior to the circle." Nice and clean, and it gives the expected results in the case of a convex polygon: a triangle has three sides, a square four, etc. And a circle has infinitely many, since every line tangent to the circle is now a "side" under this definition.

This whole question reminds me of Hardy's remark (I paraphrase) that a huge conceptual advance in mathematics is when we stopped asking what things

posted by escabeche at 6:14 AM on December 21, 2006 [4 favorites]

As many people have told you, the question is not very well-formed since you have not defined what you mean by a "side." If you mean by a "side" a line segment of positive length, then of course there are none. But this is not a great definition -- for instance, a triangle, or any other polygon, contains

*infinitely*many line segments of positive length -- in order to pick out the ones that you mean when you say "sides" you will have to make your definition more complicated.A better definition, though not the one you might first think of, is: "a side is a line which intersects the circle but does not contain any points interior to the circle." Nice and clean, and it gives the expected results in the case of a convex polygon: a triangle has three sides, a square four, etc. And a circle has infinitely many, since every line tangent to the circle is now a "side" under this definition.

This whole question reminds me of Hardy's remark (I paraphrase) that a huge conceptual advance in mathematics is when we stopped asking what things

*were*, and instead started asking what things should be*defined*to be...posted by escabeche at 6:14 AM on December 21, 2006 [4 favorites]

Reminds me of how many sides does a stop sign have? 8? 10?

posted by allkindsoftime at 8:15 AM on December 21, 2006

posted by allkindsoftime at 8:15 AM on December 21, 2006

I've always seen it as infinite, although I'd agree that it's probably definitional. But I'll throw my logic into the pile: the more sides you add (assuming they're uniformly distributed...), the closer it looks to a triangle. Going from a triangle to an octagon, it starts to look like a really blocky triangle. Check out this diagram (found randomly on Google Images). If I squint a little bit, the "undecagon" looks like a circle.

Keep adding sides, and it'll look even rounder. Keep adding sides forever, and you get infinite sides.

posted by fogster at 8:35 AM on December 21, 2006

Keep adding sides, and it'll look even rounder. Keep adding sides forever, and you get infinite sides.

posted by fogster at 8:35 AM on December 21, 2006

I think that by "sides" the questioner clearly means "lines". I'm no mathematician, but I agree with

I also agree that we are talking in terms of the Geometry that we all learned in school. This may help: Glossary.

If "side" means "curve" they you are talking something completely different (and not comparing it to shapes with line sides any longer). See: Naming and Classification of Curves

posted by spock at 8:41 AM on December 21, 2006

**Tacos Are Pretty Great**. A point is a singularity. The circle, as shown on a math book page, is a**representation**of the circle. You must ignore the apparent thinkness of the line (.etc) because it really doesn't exist (in any direction). If you take any two points on that circle (no matter how close - even "adjacent") and connect them with a line, now you have a "chord". A circle would be made up of an infinite number of chords, but definining it by its chords would make it a polygon - not a circle. Depending upon your "resolution" it may appear to be a circle, but it no longer would be (if you could look closely enough).I also agree that we are talking in terms of the Geometry that we all learned in school. This may help: Glossary.

If "side" means "curve" they you are talking something completely different (and not comparing it to shapes with line sides any longer). See: Naming and Classification of Curves

posted by spock at 8:41 AM on December 21, 2006

IAAEmbarrassedMathematician. Who will not post about math in a hurry any more. Please ignore previous comment; of course, the definition I proposed applies to infinitely many lines through any vertex of a polygon. A better definition (I think!) would be: A side is a line segment L contained in your shape, which has the property that no line segment L' containing L is contained in your shape. Under this definition, each point of the circle is indeed a side (recalling that a point is a line segment of length 0) while the sides of a triangle, are, well, just what you think.

posted by escabeche at 9:04 AM on December 21, 2006

posted by escabeche at 9:04 AM on December 21, 2006

infinite number of sides--you can draw tangents to every point on the circle

posted by uncballzer at 9:51 AM on December 21, 2006

posted by uncballzer at 9:51 AM on December 21, 2006

This is a semantic problem, not a math problem, as illustrated quite well above.

To have sides, you have to have vertices. This ends up leading you through the same rabbit hole since you have to then ask does each point of a circle represent a vertex.

escabeche is closest in trying to say a side is what connects two vertices, but is not contained within the interior of the form.

But, yet again, this falls apart due to the question of are all points of a circle vertices.

My opinion is that in all regular closed polygons, there exist points that are circumscribed by the shape, that are not vertices. Therefore, in closed regular polygons, not all points are vertices. The other way to say this is that in all regular closed polygons, there exists a finite number of vertices.

So, in a circle, which is a regular closed polygon (or is it?), all points circumscribed act as vertices. However, all points of a polygon cannot be vertices, so this means a circle has no vertices. You cannot have sides without vertices.

So, a circle has no sides.

Put another way, for a circle to be considered to be made up of all vertices, then a straight line would have to be considered to be made up of all vertices.

However, the "real" answer is the "the question is too ambiguous to be answered".

I will point out that the people concentrating on tangent lines are missing the point. There are an infinite number of tangent lines that can be drawn through/against the vertex of any polygon. For instance, in an equilateral triangle, you can draw an infinite number of lines that contain vertex A between where they overlap side A and side B.

posted by Ynoxas at 10:47 AM on December 21, 2006

To have sides, you have to have vertices. This ends up leading you through the same rabbit hole since you have to then ask does each point of a circle represent a vertex.

escabeche is closest in trying to say a side is what connects two vertices, but is not contained within the interior of the form.

But, yet again, this falls apart due to the question of are all points of a circle vertices.

My opinion is that in all regular closed polygons, there exist points that are circumscribed by the shape, that are not vertices. Therefore, in closed regular polygons, not all points are vertices. The other way to say this is that in all regular closed polygons, there exists a finite number of vertices.

So, in a circle, which is a regular closed polygon (or is it?), all points circumscribed act as vertices. However, all points of a polygon cannot be vertices, so this means a circle has no vertices. You cannot have sides without vertices.

So, a circle has no sides.

Put another way, for a circle to be considered to be made up of all vertices, then a straight line would have to be considered to be made up of all vertices.

However, the "real" answer is the "the question is too ambiguous to be answered".

I will point out that the people concentrating on tangent lines are missing the point. There are an infinite number of tangent lines that can be drawn through/against the vertex of any polygon. For instance, in an equilateral triangle, you can draw an infinite number of lines that contain vertex A between where they overlap side A and side B.

posted by Ynoxas at 10:47 AM on December 21, 2006

*Sorry, but you missed on this one. Better luck next time.*

In math discussions your argument should stand for itself. If you need to add bluster, you're likely trying to prop up an argument thats weak to begin with.

I've found this to be a good guideline in math/science ask.mefi threads.

posted by vacapinta at 11:34 AM on December 21, 2006

(Apologies for thread-jacking.)

@escabeche: I really like your definition of "side" in this case; it works for all polygons (even concave ones) in a way that seems "natural", while also giving curves infinite sides. I've never seen this definition before; did you just make it up, or is it somewhat standard?

posted by insipidia at 3:27 PM on December 21, 2006

@escabeche: I really like your definition of "side" in this case; it works for all polygons (even concave ones) in a way that seems "natural", while also giving curves infinite sides. I've never seen this definition before; did you just make it up, or is it somewhat standard?

posted by insipidia at 3:27 PM on December 21, 2006

*Reminds me of how many sides does a stop sign have? 8? 10?*

2 (front & back)

posted by flug at 6:18 PM on December 21, 2006

**insipidia**: just made it up, but presumably it is "standard," to the extent that people still think about such things at all....

posted by escabeche at 10:22 PM on December 21, 2006

This thread is closed to new comments.

... but thats just my guess.

posted by MathewS at 6:49 PM on December 20, 2006