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If you’re sitting next to a window, open it up and look out at the sun. First, you might think “Wow! It’s bright outside.” Then, the next thing that pops into your mind might be that it looks especially yellow or orange or big or small on that particular day.
Eventually, the fact that the sun is a circle may also pop into your head, and rightfully so. The circle is one of the first concepts we learn about. It’s a common shape, and we catch glimpses of the same shape every few seconds.
Sitting down next to a friend and having a chat? Chances are you’re mostly looking at your friend’s circular pupils when talking to them. Sipping cola from a straw? You’ll notice that the hole of the straw is circular in shape (you might also eventually wonder how many holes a straw has, but that’s a debate for another time). Circles are all around us.
Yet, when we think of circles, we start and stop exactly here. We think of these everyday instances of circles. We think of the Sun, we think of the straw, and we think of the pupils in our friend’s eyes. We don’t actually ever ponder what a circle is.
When I ask you this question, I’m not questioning the existence of a circle, rather, I’m questioning what we can classify it as. For instance, think of a square. We know that squares are a type of rectangle with four equal sides. They’re also parallelograms; the opposite sides of a square will never meet or intersect with one another. Finally, squares can also be considered polygons.
What is a polygon?
A polygon is any two-dimensional shape with a finite number of sides. Technically speaking, based on just the two criteria I’ve mentioned above, a circle would be a polygon; but is it, actually? Let’s find out together!
A few geometric rules
Although a circle is, in fact, a two-dimensional shape, it doesn’t really have a finite number of sides. Long story short, a circle has no sides… yet, that doesn’t make sense. A circle is a closed shape, and based on basic logic, any closed shape is enclosed by a boundary, that is, a side. So, what’s with the notion that a circle has no sides?
If we dive deeper into that story, a circle actually has an infinite number of sides to it. If we follow this train of thought, then very clearly a circle isn’t a polygon because a polygon has a finite number of sides.
If you’ve kept up with me till here, you, like me, might be confused. So far, we’ve got two completely opposing hypotheses here.
Hypothesis 1: A circle has zero sides.
Hypothesis 2: A circle has an infinite number of sides.
Both of these hypotheses give us an incomplete idea. If we go with hypothesis 1, it doesn’t make sense, as a closed shape must have at least one side.
On the other hand, when we draw a circle, we don’t freehand a circle with an infinite number of sides. So, what’s the right answer, if there is any?
To answer this, we need to wind the clock all the way back to ancient Greece, to the time of Socrates. Socrates was one of the world’s finest and most ancient thinkers. Socrates came up with many famous thoughts and creations, however, his greatest gift to mankind might be his method of inquiry.
Socrates, instead of always spouting answers, would mostly pose questions to pupils. So, instead of answering the question of whether a circle is a polygon, let’s instead counter by asking what exactly constitutes a side?
What is a side?
To ponder this, we’ll continue down the path of hypothesis two, that a circle is a polygon or at least a closed structure with an infinite number of sides. In this definition of a circle, we can consider each individual point that makes up the circle to be a side. Here, we run into another problem. If every point of the circle is a side of the circle, then very clearly, each side of the circle is placed immediately next to another side of the circle.
How is this a problem?
Because this means each point of the circle is a side, and each side of the circle is also a corner !
Another problem that we eventually end up dealing with is the length of each of these sides. In a shape with an infinite number of sides, made up of an infinite number of points, the length of each of these points/sides is undefined. In other words, it comes down to zero, which again, just doesn’t make any sense.
If there’s any conclusion to this article, it’s that you’ll be leaving with many more questions than answers (if any); yet, what we can conclusively say is that circles are just closed structures, as opposed to polygons. Although circles are closed, two-dimensional structures, no one really knows how many sides a circle has nor what category of shape it is!
So, let’s leave the bickering and the proofs to seasoned mathematicians and just appreciate circles for what they are; the Sun, the straws, and the pupils of our eyes.
Readability: 68.9
Flesch Kincaid Grade Level: 6.6
Glossary
Pupil: The circular, round portion of the eye that is responsible for the regulation of light.
Two-dimensional: Any object that has length and width but does not have any depth. For example, a circle is a two-dimensional object but a sphere is a three-dimensional object with depth.
Corner: Connects two sides together.
References
Is a Circle a Polygon? (2023, February 27). What’s Insight. https://whatsinsight.org/is-a-circle-a-polygon/
Kumar, A., & Kumar, A. (2023, March 17). What is a Polygon? Shape, Types, Formulas, Examples, Facts. SplashLearn – Math Vocabulary. https://www.splashlearn.com/math vocabulary/geometry/polygon
How Many Sides Does A Circle Have? (2019, September 6). Simple City. https://richardelwes.co.uk/2019/09/06/how-many-sides-does-a-circle-have/
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