HELLO YES I AM BACK. I spent the first couple of months this year working on my novel. But I'm officially back with new videos, more frequently, for your eyeballs and earballs and brainballs.

I'm on a "The wheel is turning the wrong way!" side.

Circle is the most neutral thing, it has never taken a side!

Me an blender user that knows that circle has as many sides as PC can handle.

Intuitively: inside + outside = 2 sides. Physically: Circle perimeter / Planck lenght = almost indefinite but still definite, much like Australia.

I love how micheal is subtly chaotic but Kevin’s just losing his mind

Opening question: "How many sides does a circle have?" 5 minutes into the video: "The coast of Australia is infinitely long" Yup, it's a Vsauce video alright.

here's more proof it could be infinite: So when pi didn't exist, people tried to measure pi by using "gons". So imagine you had a triangle and a circle. (a triangle is the "gon" with the least amount of sides.) Say you put the square inside the circle and shrunk/grew the square until only it's corners were touching the outside of the circle. This would give you the minimum of what pi could be using this technique, because a triangle has the least sides of any polygon. If you put the circle inside the triangle, this would give you the maximum. People did this with polygons with HEEAAPS of sides, until the new formula was discovered. Using the old method, you could only get a finite number for pi. But if you kept doing this infinitely, you would get a circle, and infinite digits of pi. So basically, a polygon with infinite sides equals a circle.

Ive always accepted the fact that a circle has one side. It’s definitely a very very interesting topic to dive into, but as said in the end of the video, it’s up to your own interpretation. I like to believe a circle is one side without any edges, bent together to make fit.

This feels like one of these classic Vsauce videos, like "Is Cereal Soup?". Asking the important questions. I like it.

"If you have an infinitely small measuring stick, the shoreline would be infinitely long." **tl;dr** Calculus says otherwise. Using increasingly smaller measuring sticks until those measuring sticks are infinitesimally small is the same as taking the limit of a function. And the result is often *not* infinity but rather an exact, finite value. You can take the limit of a function as x approaches infinity and often find that the end answer, where x = ∞, is *not* itself infinity. As an example that's highly relevant, you can estimate the circumference of a circle by inscribing polygons inside of that circle and measuring the perimeter of that polygon. First you start with an equilateral triangle inside the circle, and the triangle's perimeter will give you a (rough) estimate of the circle's circumference. Then you move to a square, and its perimeter will be closer. Once you reach a decagon, you've got a decent estimate of the circle's circumference though still not an exact match. By your logic, if we increased the number of sides of our inscribed polygon to infinity, thus reducing the individual length of each side to an infinitesimally small point, we should get an infinitely large circumference. But that's not what happens. The formula for the perimeter of a polygon inscribed in a circle is 2nr*sin(π/n) where r = radius and n = number of sides of the polygon. e.g. to find the perimeter of our initial triangle for a circle with a radius of 6 (and thus a perimeter of 2πr ≈ 37.7), we would use 2*3*6*sin(π/3) which is about 31.18. Not too shabby for a rough estimate using only 3 sides. If we bump that up to a decagon, we get 2*10*6*sin(π/10) or 37.08. Now we're *really* close. If we use 1,000 sides, that's 2*1000*6*sin(π/1000) or 37.699. Notice that we're using significantly more sides, meaning the length of each side is increasingly approaching an infinitesimally small value, and yet our result isn't approaching infinity. It's approaching the exact circumference of our circle. And sure enough, if we use calculus to take the limit of our formula as n approaches ∞, we get exactly the circumference of our circle. As your measuring sticks become smaller, the result doesn't tend towards infinity. It tends towards a finite, exact value.

"How many sides does a circle have?" It's easy, it's either 0-n

" A circle has one side. Or does it...?" *Vsauce music starts playing*

One face, infinite sides: the story of a monogon looking for Australia

We ain’t talking about how the wheel was rolling to the left and the screen was moving right

When I first learned what a circle was when I was 2, I had no idea how complicated it was.

Q: How Many Sides Does A Circle Have? A: All of them

4:11 Huh, that’s an interesting way a wheel spins when rolled on the ground

I feel like rather than saying lowering the unit of measurement increases the size of the Australian infinitely is a bit much, rather that it approaches its true length forever but never quite reaches it, like an asymptote graph.

I like to imagine circles as the limit of the series Pn, when Pn is a regular polygon with n vertexes, with a fixed distance from a certain point. As n approaches infinity, the number of sides approaches infinity. So saying it has infinite sides makes more sense to me

Only SpongeBob is capable of drawing a perfect circle

However many sides a circle has, the logic behind it can’t suggest that a square has more or less than four sides.

I’ve always thought that the concept of a circle has infinite sides, but an actual circle you see has half as many sides as there are pixels/atoms around the edge

“A circle has two sides: inside and outside” -Knuckles in Sonic Boom

I always thought that the most-agreeable answer was that a circle has an infinite number of sides. For instance, if one thinks about regular polygons (those with equivalent side lengths & internal angles), adding more sides will cause the shape to become more and more round. So, in that sense, a circle could be described as a regular polygon whose side count approaches infinity.

Fun Fact: a polygon with an infinite amount of sides is called an “apeirogon,” which is - strictly speaking - not a circle.

Thanks for the video. I like how you manage to present simple math in a way it becomes interesting. However, let me disagree with you on one of the points of the video. The coastline of Australia is finite in length. The process you described is very similar to Zeno paradox (the one with the Achilles and a tortoise). Even though it remained sort of unanswered for a long time, after differential calculus was invented, this paradox lost it's basis. The process of dividing the coastline will bring you to the Darboux sum, which in the limit of infinitely small segment, leaves you with the integral equal to the length of the coastline. Simply, when the length of a segment decreases their number increases in a way that the total length remains finite.

Ok, here's my attempt at a rigorous answer to the question. Basically the question boils down to "what defines a side?" There are clearly many different interpretations and definitions, but we know that an n-gon should always have n sides. We would also expect that deforming a side of an n-gon in some continuous way would keep the number of sides the same (like if we made a side of a triangle wiggle a bit or whatever). Lastly, we know that all polygons can be thought of as paths through space, and that there are corners at the ends of sides. With that, let us define a shape as a continuous map γ: [0, 1] -> R^2, in which γ(0) = γ(1). This defines a closed loop in 2-space. Notice that, for a polygon, dγ/dt is discontinuous if and only if we are at a corner. Hence, we may identify the corners with the discontinuities of the derivative - except if we start at a corner. To fix this, we define the domain of the map to instead be R/Z. This way, we identify γ(1.1) with γ(0.1), and continuity can still be defined at 0. As such, for a shape γ: R/Z -> R^2, the number of corners of γ is the number of discontinuities the derivative has. Then, we can define the number of sides as the number of maximal intervals over R for which γ is continuous. This makes sense for polygons, as we made sure it had to. For a circle, however, there is only one maximal interval of R for which γ is continuous, namely R. Hence, by this definition, the circle has one side. I understand that this definition is not really a proof, but it does seem like the most mathematically natural one to me. I simply can't see a consistent definition in which a circle would have no sides or two sides. I can imagine there being infinite sides, though, if we only let polygons have straight sides, and then letting a circle be a limiting polygon. However, I don't know if that's useful, since the deformation property I brought up no longer holds.

Kevin you're our last Vsauce hope. This video feels like a back to the roots thing. Can't wait for more

This is the sorta voting I like, getting a nation to decide how many angles a circle has

About the coastline of Australia, I feel like we can acknowledge that while you can divide any finite distance into infinitely many smaller points, that doesn't mean that the distance is longer. Like how as you look at an object closer and closer it appears bigger, but it's not really.

I wouldn't say the coast line of Australia is infinite in length. Sure, you could reduce the size of the intervals for measurement and the length keeps increasing, but it will increase by smaller and smaller amounts. It' like doing a numerical integration in Calculus, the value will eventually converge to a finite number. Still, awesome and thoughtful video as always :)

When I was in second grade, the teacher shamed me in front of the class for saying a circle has an infinite number of sides. She said a circle has zero sides. I was very young when I realized people could obtain power without sense.

“degenerate polygon” Idk why but I strongly identify with this phrase

5:03 i love that you made yourself laugh, this is hilarious xD

"How many sides does a circle have?" "Screw it all the shapes have infinite sides" "No I mean the circle-" "ALL OF THEM HAVE INFINITE SIDES OK?"

he never explained why it was so critical for Australia to learn how many sides a circle has

Why yes, I do call circles "infinigons." How could you tell?

Funnily I still remember that I made the same argument to my maths teacher back in high school several decades ago. She dismissed the idea that a circle has infinite sides just in a split second, which left me quite baffled. Thank you for showing me this, this gives me some kind of peace 😂

Technically, the answer would be different based on how big the circle is, whether it is filled, or not; what the circle I saw made out of, it all has to do with the atom size, and how many there are. Every circle does indeed have a define(d, able) number of sides.

0 sides or sth idk smh

I've always been a believer of a circle having infinite sides

I think it is infinity because if a cube is pushed using its top, it will roll and get slower every time it falls on a side and how much it gets slower depends on how big the side is compared to the object. But when it's a ball, it will never slow down by falling on its side as the sides are much smaller than the object

4:09 the wheel spinning the wrong way is the bane of my existence

The real answer to this question is another question: “In what context?”

a running gag with friends is that it has two sides: inside and outside. rather stupid, but the reactions are almost always precious.

I am part of the infinite side team. I am since my elementary teacher tought us how to calculate the area of a circle, considering the circle as a regular polygon and the area of it as (Perimeter * radius) / 2 (that becomes pi * square radius)

It depends on if you quantify sidedness based on continuity or linearity. You could argue that a circle has infinitely many tangents so it has infinitely many sides. Think sort of how Archimedes used increasingly higher sided ngons to make a circle But then you could realize that if it had infinite sides it would require infinite angles, and all 2d convex shapes in Euclidean geometry have internal angles of

If we define a side of a two-dimensional object as a straight line of any distance greater than zero between two angles, I would argue that a circle has no sides; but has infinite points.

Hey Kevin. I just wanted to point out something I thought was cool with the real life vs pure mathematics. Technically the Planck length is the smallest distance in the universe that anything can happen. Therefore Australia can’t have an infinite coastline because the measurement cannot be infinitely small. I love this fact because it completely breaks down thought problems like variations of the Achilles Race problem and other supertasks

“How many sides does a circle have” Kevin- WRONG!!!

Woah, this video came out on this day, exactly one year ago! That's crazy! Also, I like to say that a circle has 2 sides, inside and outside.

4:11 That wheel is spinning in the wrong direction

Who else wishes he'd link to an article about why Australia needed to know that?

I would like to make the two cases. One: all you are really saying at the end is it's about context, and I'm sure the answer they were looking for in Australia was One(1), as in the number of side/angles. And Two: The reason I'd answer as such is because you could see a circle as if it's a straight line that is bent (not angled) to connect to itself at a 0 (or 360) degree angle, but because it happens at a singular "point", you can't actually see the angle at our level (or possibly any level) of measuring standards. But... it is one side and one angle, in that context, which is what most mean when they are talking about it. Grasping at straws may not be lazy, but it doesn't mean it's right, so the obvious is usually the best. ::cough cough razor::

“A side is, a side.” Hm, yes. The floor is made out of floor.

4:13 The wheel is spining the wrong way. :/ ili

It may be hard to decide how many sides a circle has but I can promise you there is no downside to subscribing to Vsauce 2.

All these squares make a circle. All these squares make a circle.

It has infinite sides. When you keep adding sides to a polygon, it becomes more and more like a circle. So after infinity sides, it would be a circle.

*Infinity* "I see, this human doesnt know me." Googol plex: forgive him my lord G64: nah, he's drunk Rayo's number: we'll teach him a lesson my lord

“The coast of Australia is infinitely long.” Me: Or it’s just a heck of a lot of Planck lengths

You can define A(x) being a function mapping an integer greater than two to the area of the surface of a regular polygon of x sides inscribed in a unit circle. You can define P(x) being a function mapping an integer greater than two to the perimeter length of a regular polygon of x sides inscribed in a unit circle. Both A(x) and P(x) converge respectively to the length of the circle and the area of the disc. It begs an interesting question: What about regular polygons with a non integral number of sides ? If we look at the perimeter and area function, we want continuity and order. For that, we can start inscribing the n-1 sides, the n-1-th side would have it's length at prorata, the n-th length connect the two open vertices. So, a 3.5 regular polygon will have half a square caped by a right triangle. Two sides of sqrt(2), one of sqrt(2)/2 and the last side of sqrt(3)/sqrt(2). A geometric proof is possible. If you free one vertex in a regular polygon, that vertex and the two neighbors defines a triangle. Let call the vertices A for the freed one and B and C for the neighbors. The area of that triangle reach a maximum when A is further away from BC. Same for the perimeter, minimal length is |BC|, max length is n|BC| and it's continuous.

Monagon is now my new favorite super hero and Zilchagon is more my new favorite word.

When humans create the definition of something, then argue the meaning of the definition. Just chasing our own tails.

Basically this video: how many sides does a circle have? Kevin: *YES*

I think it should be 1. That way, you might be able to classify a straight line as having 0, smooth curves as having >0,1,2,

My approach would be... A perfect circle as a 2D spacial idea would have an inside and an outside. So no angles for the perfect circle for it is an idea without angles and not a physical object.

Many years ago my 9th grade math teacher asked the class how many sides a circle had and I answered "one side" and the entire class laughed at me and then the teacher said no there's 0 sides and moved on and I am still confused why I was laughed at to this very day I don't understand.

I was literally just thinking about this this morning. I found a more interesting question was how many sides dose a semi-circle have.

One. It's just one side rounded into, well, a circular shape. There is such thing as a bigon or digon, which is a 2-sided shape that is shaped similarly to a human eye. In order for it to be a polygon with two sides but not just have its two sides in exactly the same location as each other if its sides had to be limited to being straight and not curved, it has to have its sides curved a bit, excluding semicircles though, since they have one flat side and the other side just takes on a circle, but obviously the still-existing portion of it, assuming that the way the semicircle in question was created by cutting a circle into two parts and discarding its other part. Therefore, according to my definition, a circle has one side.

To the coast part You said if you keep getting smaler massermants then the coast will get longer, but if we imagin the smallest thing (Atom, Electron or something smaler) is exactly one flurp (totally made up mesering unit) long, wouldnt that be the maxed lenght? Because if the would be some smaler messurment Unit, maby a gazorp (again made up) wich is half of an flurp it, the coast still would stay the same lenght? Pls Proof me wrong if im wrong

5:00 ive never seen kevin laugh before tbh

For the “Australia is infinitely long” analogy, wouldn’t the increase in distance diminish as you go to smaller and smaller units? Then the measurement of Australia’s coastline wouldn’t go to infinity, but rather converge to a number… right? It’s been 8 yrs since I’ve taken math, excuse my rusty knowledge

Kevin: Australia has an infinitely long coastline Planck length: I'm about to end this man's whole career

Infinite sides is what makes most sense to me. This lines up with size of each side and its relationship to the number of sides. If the number of sides increasing causes correspondingly decreasing proportions of each side, then it makes sense that a polygon with infinite sides has each side occupying 0% proportion. It's strange to rethink the line segment as being something that has a start point that overlaps on its own endpoint, but that kind of counterintuition seems at least characteristic of what we can expect when dealing with limits. Edit: choosing the 0 sides team makes even less sense in a way because a polygon that undergoes a progressive upscaling of its sides would somehow "return" to zero instead of the upper limit. What makes more (intuitive) sense?: Number of sides: (3,4,5,6,7,... infinite) or Number of sides: (3,4,5,6,7,...0) Though both the infinite sides team and zero sides team have equivalent arguments, only the former satisfies the intuitivep pattern as we increase sides. I think this question should have us revise the definition of what minimally qualifies as a line segment. Maybe a "0 dimensional line" can be quantitatively equivalent to a point while still being semantically a different concept.

My sadness when he didn't end the video with "I will see you on the flipside"

2 sides makes no sense. By that logic a triangle could only have 2 sides (inside and outside) or 6 sides (both sides of each of the three sides). Counting the inside of shape isn't in the spirit of countirng sides.

I want to make two points about "how does the coast line not become infinitely large?", since I don't think your conclusion is correct in this generality. 1: Consider a circle. Its curvelength can be measured by the arclength formula (measuring with infinitessimal rulers leads to an integral), integrating from 0 to pi. The length of this curve is finite. 2. More simply put, other infinite series can also converge, even though their partial sums are strictly increasing in size. Directly related to this would be the convergent series $\sum_{i=1}^\infty n/2^n$ that kind of emulates an increasing measurement, while having a finer and finer measurement.

well it depends, if its digital and you can zoom infinitely it has infinite sides but if its physical(in real life) it depends on how big the circle is because if you count how many atoms is on the outer most layer that's how many positions it can rest on but it would have to be completely inflexible(witch is impossible) so there is no single answer

Imagine saying circle when someone asks for a shape with one side This post was made by Möbius strip gang

kevin: australias coast line is infinitely long planck length: am i a joke to you?

Hi Kevin! Fellow mathematician here who love your videos. 2 sides is not an option as it would mean a triangle has 6 sides and a square 8. Now if we consider paradoxes in general, many of them come from an ambiguity in definitions. Common language is a real danger in maths. For this one we should give a clear definition of sides to avoid the paradox. - If it is the number of line segments forming a shape, then the circle has none. - If it's the minimal number of sets of points aligned, then the circle has an infinity of them. - if it's the number+1 of changes of a kind of derivative then it could be 1. Hope I'm clear, English is not my first language. :-)

Seriously this whole thing with two sides is the best out"side" of the box way of thinking about this I ever heard

In blender the circle has 32 sides and it looks pretty round to me

I honestly thought this was a pretty strait forward question. A circle has one side, as defined by it's interior angle, and by the "outside" versus "Inside" definition. Inside/outside is not a side, its a position, relative to the side already defined. That is akin to calling beside a side. I can beside a side in many different positions, but that doesn't make all those possible beside positions each a different side. A Side should not be defined by a point, because then you run into all those problems with out to define the point. Defining a side by a point causes all sorts of problems for any curve or non-flat face.

I knew it was 0 , but when Vsause made this video,I begun questioning my existance

As an Australian, I remember googling how many sides a circle has

Personally I'd think a circle as a "limit" polygon, ie. limit as the number of sides approaches to infinity -> Infinite sided polygon.

I'm still in so much love with this series Kevin, thanks for being you!

i like this. every shape has two sides in the same way, inside and outside. perfect.

Addressing units of measurement. If we use the smallest possible measurement of length, the planck length, we can determine how many sides a circle has. Of course if the circle has a diameter of a planck length, we would need to go even smaller. But if we drew out a circle of a nanometre, or a centimetre, and so on, we may get to a more discernable conclusion. It all depends on what that specific unit of measurement is that is relative to that point of contact between that circle and a flat surface. Use that point of measurement and fill in the rest of the circumcerence with equally small points and you have your theoretical answer. Easier said than done. I am sure a mathematician could write out an equation where p=point of contact, c=circumference, tje diameter, and whatever else.variables required for the equation.

Saw the title. Paused it at 0:02 just to write “this is gonna be one of those vsauce videos that breaks my brain”. Back to our regularly scheduled programming

Infinite because the more sides a shape has, the closer it gets to being a circle, some shapes are even indistinguishable from circles

A circle wouldn’t be a none-ogon, it would be a sides went-gone

Next time someone doesn't believe me: "You SHOULD!"

4:11 it's... spinning the wrong way

On an episode of the Batman TV series, the Riddler posed the same question: how many sides does a circle have? Robin answered the question. Two: inside and out.

It depends on how you label a side but one way you could do it is drawing it on a graph and saying that any given point on a certain side will have a tangent that shares the same equation with the tangent of a different point on the same side and by this logic the circle has an infinite number of sides as no 2 points on the circle will share an equation for the tangent

Breh why that wheel moonwalking XD 4:10