The Instigator
Con (against)
4 Points
The Contender
Pro (for)
0 Points

There are 360 degrees in a circle

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Voting Style: Open Point System: 7 Point
Started: 7/9/2015 Category: Education
Updated: 1 year ago Status: Post Voting Period
Viewed: 1,159 times Debate No: 77457
Debate Rounds (5)
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Votes (1)




It appears people have accepted there are 360 degrees in a circle solely because the Babylonians believed this.

I am hoping to find an opponent who can convince me that there are in fact 360 degrees in a circle

I will begin:

A straight angle has 180 degrees, therefore:

There are 180 degrees in a 3 sided shape (triangle) because the sum of it's interior angles equal 180
There are 360 degrees in a 4 sided shape (square) because the sum of it's interior angles equal 360
There are 540 degrees n a 5 sided shape (pentagon) because the sum of it's interior angles equal 540
There are 720 degrees in a 6 sided shape (hexagon) because the sum of it's interior angles equal 720


There are 180 degrees in a 1 sided shape (circle) because the sum of it's interior angles equal 180.

Good luck Pro



I thank @mostlogical for creating this debate, and I accept. BoP is fully on CON as he is not only the instigator but also proving something a majority don't believe in. Despite this, I feel that I should point something quite obvious out to the audience:

A degree is physically based off of the circle. The Babylonians created the "degree" several thousand years ago, and, since then, a single degree was always 1/360th of a full circle. So, as logic hints, there are exactly 360 degrees in a circle.
The number 360 was probably taken from the Earth's revolution around the sun. The Babylonians discovered and logged the changes that different seasons bring, and estimated that a year is 360 days. They weren't that far off, only 6.25 days.


I will anticipate CON's points, or rather explanation, for how a circle could possibly equal only 180 degrees. Just to help my partner out, I might add that the 180 degrees for every angle rule counts only for polygons, while a circle is technically not a real polygon.

Awaiting further posts...
Debate Round No. 1


Thank you for being open, and providing a great introduction

Below is the maths I used to calculate the angle for a circle:

16 sides:
360/16 = 22.5 degrees
There are 180 degrees in a triangle, so the interior angles add up to 180 - 22.5 = 157.5

1000 sides:
360/1000 = 0.36
There are 180 degrees in a triangle, so the interior angles add up to 180 - 0.36 = 179.64

3 million sides:
360/3 million = 0.00012
There are 180 degrees in a triangle, so the interior angles add up to 180 - 0.00012 = 179.99988

20 million sides:
360/20 million = 0.000018
There are 180 degrees in a triangle, so the interior angles add up to 180 - 0.000018 = 179.999982

If I continue increasing the number of sides of a polygon by repeating this method above with ever greater sides, each interior angle will become "flatter" i.e. closer to 180 but will NEVER reach 180 degrees. This is because finite polygon lines are not curved and a circle does have one side like you mention. However when there are an infinite number of sides it is clear the interior angle will be 179.9... or in other words 180, and the sum of it's angles will not be infinitely high because there will be an infinite number of interior angles perpendicular to each other, creating one angle.

A circle and a polygon with infinite sides are therefore the same thing. If this was not true then the method used to calculate pi would not be entirely accurate, yet it is, and the circumference can be calculated despite it being a curved line. The Babylonians couldn't calculate Pi accurate to two decimal places so they had no idea how to calculate the angle of a circle. If they could then I doubt they would have imagined the central angle to be a circle which seems to be the case. Whatever the reason for them choosing 360 we have accepted it without applying maths.

Mathematicians do not show angles in more than 1 decimal place, so this means they use a polygon with 647,640 degrees to make angle measurements. Therefore there is no need to "make up" the number of degrees in a circle. This is why it doesn't make sense to me. People do often say angles are about rotation, and I will leave it to you to prove this point, or any others you wish to share

Many thanks


Several misconceptions I see here, but geometry isn't my strong suite so feel free to correct me.

1. In your beginning equations, I do not understand why you keep repeating "there are 180 degrees in a triangle". Once we start talking about polygons with any different than three sides, it suddenly stops being a triangle and becomes something else. Instead you should say "there are 180 degrees in a strait angle", as an interior and exterior angle combined make a strait angle.

2. I don't understand this as well. In the first round and even here, you agreed with me that a circle has only one side, but now you say it has an infinite amount of sides. Please choose one and follow it.

In reality, a circle is a curve, and there is only one. Therefore, not only can you not refer to it as a polygon, but it also has zero actual angles. Instead, a circle is measured by the rotation of the one curve, which equals 360.

3. "there will be an infinite number of interior angles perpendicular to each other, creating one angle"
There's two things I find confusing in this statement.

(A) I don't even understand it. Angles can't be perpendicular to each other, only lines can. If there is a way to make angles look perpendicular, please attach a picture to explain. I think I bent my brain trying to illustrate it.

(B) Maybe perpendicular angles would cancel themselves out if they existed? Or would they combine into 180 degrees?

4. I didn't even think of Pi for a moment during this entire proof until you brought it up. One does not need Pi to understand this, so the Babylonians could have understood it as well with a primitive understanding of Pi as well.

5. IF there is an infinite number of sides, each angle would be exactly 180. If it were 179.9, it wouldn't be infinite, because there is a slight tilt after each segment, which in turn would eventually lead it around and back into it's end, making a finite, though very large, polygon.
Debate Round No. 2


1. I did 360 minus the number of sides to determine the exterior angles. Then 180 minus the exterior angle to determine the interior angles.

I can't remember why I imagined a triangle instead of a straight angle, but I can assure you the answers are correct. There is another way of finding the angles. The way you may be taught is to find the sum of interior angles first by using the equation 180 x (n - 2) where n = number of sides, then divide this answer by the number of sides to find the interior angles.

e.g. the sum of interior angles of a pentagon (5 sided shape) is 180 x (5 -2) = 540 degrees.
Each interior angle will equal 540/5 = 108.

The more sides a polygon has the larger the interior angles are and closer they will be to 180 degrees. The difference between interior angles of two polygons with more sides is smaller than the difference between the interior angles of two polygons with less sides

2. If you imagine a point on each corner of a polygon, the total number of points is equal to the number of sides that polygon has. There are an infinite number of points on a curved line, this tells you that a shape with infinite sides is a circle.

The definition of a polygon is a figure with 3 or more "straight" sides. Circles have infinitately more "straight" sides than 3 so I see no reason why circles can't be called polygons.

My opponent claims "there are zero angles in a circle", however I have shown there are 179.9 reaccuring degrees (180) in a circle. It is even possible to gain a reasonablely accurate approximation of the angle using a protractor like it is possible to estimate the length of the curved circumference using a straight ruler. Just make measurements on a very large circle.

3. A) I do apologise for bending your mind, I've used "perpendicular" when really I mean "aligned", sorry I got mixed up. A perpendicular line creates a 90 degree angle. Adding a side to a polygon with 4 sides will create an obtuse angle which will get bigger when more sides are added. If there are a finite number of sides the line hitting a straight line will never be aligned, it can only align if there are an infinite number of sides in a shape.

3. B) Whenever two straight lines meet the sum of the interior angles remains the same i.e. 180 degrees. This is true for curved lines too and a curved angle is the same no matter how much it is curved.

4. Here is the method to calculate Pi:

The point I was trying to make here is that the Babylonians probably wouldn't have even imagined that a circle can be made by increasing the number of sides a shape has. Thinking about it now, it wasn't worth mentioning. This argument can be dropped.

5. A side and a point are effectively the same thing, if there can be an infinite number of points there can be an infinite number of sides. I put three dots after the last digit of 179.9 to show there are an infinite number of 9's repeating after the 9 as I don't know how to put a dash line on top of a digit. This argument can be dropped.


Your logic seems very sound, but there is definately a problem with changing a "curve" into an "infinite series of lines". I can't put my finger on the exact reason though, you'll have to figure that out yourself.

Anyway, I have drawn a picture for you:

This is what I understand your logic to lead to. Please correct me if I'm wrong.

A short description in case the photo decides that it hates me and won't upload:

1: You say that any circleis an infinite series of connected lines. Let's presume this is truth.

2. If all the lines are connected, there must be an equal amount of angles as lines. So, an infinite amount of angles as well.

3. IF a curve is an infinite series of lines, then each angle in between has to have exactly 180 degrees. If it has any less, then there cannot be an infinte series of lines.

4. However, if each angle is 180 degrees, it cannot meet up with the other end (unless you bend space or it repeats), making a line.

Even if space is a sphere and the two ends of line line meet up, normal sized circles would be impossible.
Debate Round No. 3


I must say I do like your diagrams. It seems you are having some trouble picturing infinity, so I will try my best to help you out.

Below is a diagram I quickly made, you will notice there are 4 black straight lines and 4 straight red lines. The centres of both lines are an equal distance away from the centre of the circle, and are shown by a big red dot. Green arrows show the distance from the centre of the circle. These green distances are smaller than the ones shown by a blue arrow and this is because the blue arrows show the distance from the ends of the black lines to the centre of whatever sided shape I draw.

I believe you are struggling to fathom infinity due to picturing only the red lines which form a polygon.

In diagram 2 I have added 4 more lines/sides creating an octagon. At this point you may still be thinking that if I continue adding more sides the infinite sided polygon will not ever be a circle i.e. have one curved line, lets see.

An infinite sided polygon has infinite sides, this means I can place lines EVERYWHERE provided the centre of the line is an equal distance away from the centre of my polygon, and the distance from the ends of the black lines maintain the distance shown by the blue arrows in diagram 1. If I add any more black lines of the same length keeping the blue arrow and green arrow lengths the same as in diagram 1, the ends of the black lines must touch the outer black circle.

Adding an infinite number of sides/lines will cause a circular area to become filled. The polygon with infinite sides will just be a circle, which I have shown in blue. It is made up of the centres (big red dots) of the black lines which there are an infinite number of.

Why does a circle have 180 degrees, and not 360 degrees?

I could just say that a curved and straight line both have one side, but I'm sure that is not going to be clear enough.

I think people mistake a circle as having 360 degrees because a semi circle and a triangle both have 180 degrees just like the straight angle does so they must think that if you add them both together like adding two triangles you get 360 degrees. However if you look below at the diagram I've quickly drawn, a circle has the same number of degrees as a semi circle i.e. 180 (2x 90 degrees). If you are wondering why there are two angles in a semi-circle it is because the number of interior angles = the number of sides.

Adding two triangles toegther creates an additional side which means it gains an additional 180 degrees.

The equation for this is 180 x (4 -2) = 360.

But, when you add two semi circles toegther it does not create additional sides, therefore a circle can't have 360 degrees.



mfigurski80 forfeited this round.
Debate Round No. 4


We have accepted there are 360 degrees in a circle purely because the Babylonians thought there are 360 degrees in a circle despite them making no calculations whatsoever, and there being no genuine reason to think this today.

The only way it can make sense for a circle to have 360 degrees is if:
a) circles do not have an internal angle, or
b) the internal angle of a circle is 360 degrees.

So here is why some people think there are 360 degrees in a circle:

a) A circle is not a polygon, therefore it has no angles
b) The number of degrees in a circle (or central angle) could have been anything but 360 was chosen

The number of degrees a shape has is equal to the sum of its interior angles. If I look at a regular polygon with 3 million sides, each interior angle equals 179.99988, meaning the sum of it's interior angles = (179.99988 x 3 million) = 539999640 degrees. Increasing the number of sides will increase the sum of interior angles. By increasing the number of sides I can establish the 4th decimal point of an infinite sided polygon, and the one after that and so on, each decimal point is found to be a 9 followed by another 9. When there is a flat angle 179.9...(or 180) and an infinite number of angles of 180 degrees, there are no other sides! This is why the sum of interior angles would be 179.9... or 180 degrees, and why a circle has 179.9... degrees, rather than 360 degrees. So an infinite sided polygon is a circle.

A circle is defined as being a 360th part of a circle. This definition has been used ever since Babylonians imagined the central angle to be a circle most probably due to being unable to calculate the internal angle of a circle. Imagination doesn't make something a fact. If they imagined the central angle to be an octagon and chose to define an octagon as being a 360th part of an octagon it wouldn't mean an octagon has 360 degrees when they have defined a straight angle as having 180 degrees.

Below is proof that a triangle has 180 degrees, and since polygons are made up of triangles the sum of their interior angles can easily be calculated.

I'd like to finish by saying that the central angle is not actaully a shape, you could put a line through any polygon and say it has 360 degrees but you would only be correct in saying this if you put a line through a square. People are taught from a young age to draw a small arc around angles so when they put them together they imagine a circle has 360 degrees. Perhaps you may think it is useful however there is no good reason to make up the number of degrees in a circle. We don't even use a circle to make angle measurements!

Many thanks for reading



mfigurski80 forfeited this round.
Debate Round No. 5
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1 votes has been placed for this debate.
Vote Placed by 9spaceking 1 year ago
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Total points awarded:40 
Reasons for voting decision: ff. What a shame.