There are 360 degrees in a circle.
Debate Rounds (5)
1. Shared BoP. This is a mathematical concept, the likes of which can be proven outright or dis-proven through counter-example. Additionally, since it's accepted now in the status quo that a circle does indeed have 360 degrees, this offer of shared BoP seems more than reasonable.
2. No forfeits. Srsly.
3. No semantics. I,e. No arguing "but it doesn't have 360 degrees, it has 2pi radians...", these are obviously equivalent measurements.
I gladly accept, and thank you for reading my earlier debates on this. No-one has yet suceeded at convincing me that there are 360 degrees in a circle as taught in schools and books. I have a feeling this will be a great debate.
Good luck Pro!
Great! Thank you for accepting. Let's just right into it.
Since this is the first argumentation round, I'll try to keep it short, simply, and neat. After reading my opponent's previous debates, it is clear that he has a fundamental misunderstanding of what a "degree" actually is. Some time ago, it was agreed that it is useful to slice a circle into 360 equal parts and each of these parts was called a degree.
Now, the opponenent's argument will focus heavily on using other shapes to justify that the circle should contain 180 degrees. However, this is backwards logic, considering that the amount of degrees contained in other shapes is derived directly from the circle. Here is a visual example: https://flic.kr...
In this picture, which I'm hoping works, you can see a square on the left. Each of its angles is color coded, each is 90 degrees a piece, and together they add up to 360 degrees. By cutting up the square and reassembling its part ontop of the origin of a circle, you can see that indeed, the square perfectly breaks up the circle into 4 equal parts, each of 90 degrees.
This is how angles in polygons are measured. Given a particular angle, you place the vertex of that angle on top of the origin of a circle, then measure what portion of the circle that angle "covers". If you were to cut up a triangle in the same way we did the square, you'd see that the angles covered exactly half of the span of the circle, or 180 degrees.
For now this should suffice as my argument. In summary, the amount of degrees in a circle is a fairly arbitrary value that was set long ago and is widely used and accepted today. Any angle in a polygon is directly derived from the amount of degrees in a circle and is how many degrees the angle "spans" within a circle.
Thanks for reading.
I think I understand what you are saying, but correct me if I'm wrong. I will mainly use pictures for this round to give you a clearer idea why I think there are 180 degrees in a circle hopefully.
if you look at the below image for a moment you can see polygons are made up of triangles (shaded lightly). The sum of the angles closest to the the origin (centre) of the circle will add up to 360 degrees e.g. 120 x 3 = 360, 4 x 90 = 360, 5 x 72 = 360, 6 x 60 = 360 etc
Below I have taken your image and edited it slightly to try and make it clearer, it now appears like the square in the circle above (if you turn your head).
However I'm not sure how the above pictures prove a circle has 360 degrees. The images only show that if you draw a line (or a dot) in the middle of a circle the central angle is 360 degrees, which is true for every polygon; not just circles.
I want to show you what the central angle looks like, see below. Note there is no shape.
Since a square has 360 degrees (2 x 180) it makes sense to imagine the central angle as a square, see below
It was decided a long time ago a "point" has 360 degrees. However a point is not necessarily a circle, it can be any polygon. And you can make a complete turn by walking in one direction along the sides of ANY polygon.
We know a triangle has 180 degrees because if you see below, you will see angles a,b, c = 180 due to being below the line, and they
are all located inside a triangle, meaning a triangle has 180 degrees.
It is very easy to work out the number of degrees in every other polygon by applying the following equation:
(180 - (360/ number of sides)) x number of sides
We all know a circle has 1 side, yet if you use the equation to work out the number of degrees in a circle you will get - 180.
If you apply this for semi circles which have 2 sides the answer will be 0
These answers are wrong of course.
A semi-circle has 180 degrees like a triangle, your calculator will say 0 because a semi circle has one side less than a triangle so you are doing 180 -180 = 0. A circle has 2 sides less than a triangle so you are doing 180 - (2 x 180) = - 180
A lot of people think that if you join two semi-circles together to create a circle the angle of a circle will be 2 x 180 degrees. But it isn't like joining two triangles together. If you join two triangles together you GAIN a side and therefore add 180 degrees. An octagon has 5 more sides than a triangle so it has 180 + (5 x 180) = 1080 degrees.
If you join two semi-circles together you LOSE a side.
The best way to show why a circle has 180 degrees and not 360 is with a simple picture. I believe that if you think about the picture below it ill be clear why there are 179.9... degrees or 180 degrees in a circle.
Cobalt forfeited this round.
Here is a youtube link showing three "theories" why there are 360 degrees in a circle.
No-one knows exactly why a circle has 360 degrees because this wasn't determined by maths. Therefore my opponents argument will probably heavily rely on pictures.
I have simplified the formula to calculate the number of degrees in a shape to: (180 x no. of sides) - 360.
Apparently a circle isn't a polygon, so lets assume curved lines do not count as sides, using the formula to find out the number of degrees in a semi-circle and in a circle you will get the following answers:
Semi-circle: (180 x 1) - 360 = MINUS 180 degrees
Circle: (180 x 0) - 360 = MINUS 360 degrees
Since degrees are a POSITIVE measurement, those answers are incorrect.
People simply assume a circle has 360 degrees. One thing we can know for certain is that a straight angle measures 180 degrees, see below:
A semi-circle has 180 degrees (2x 90 degrees) People often imagine that joining 2 of these shapes together will produce a circle with 360 degrees and think it is basic maths. But below I show that joining two triangles together does not necessarily change the angle even though the shape (but not sides) has changed.
I have done the same thing again below but this time with 2 squares.
A rectangle doesn't have 2 x 360 degrees i.e. 720 degrees, it has 360 degrees just like a square. A simple way of thinking why is because the number of sides has remained the same. Joining two semi-circles together does change the number of sides but the total number of degrees does not change because an additional side is not created by joining 2 semi-circles together.
I am now going to calculate the degrees in the corner of each of these polygons (the sum is calculated by multiplying the answers by the no. of sides)
180 - 360/16) = 157.5 degrees
180 - (360/1000) = 179.64 degrees
3 million sides:
180 - (360/3 million) = 179.99988 degrees
20 million sides:
180 - (360/20 million) = 179.999982 degrees
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.
To find the angle of an infinite sided polygon i.e. a circle, we can establish each decimal point. It is clear that the larger the number of sides the more nines will follow, so it will be 179.9... or in other words 180. The sum of it's angles will not be infinitely high either because there will be an infinite number of interior angles beside each other creating one straight angle.
I look forward to your next argument
Thanks for the response and, as I've stated in the comments, I do apologize for missing the last round. However, since I made this 5 rounds long I should have ample time to get my point across.
Why 360 degrees?
First, let's address the issue brought up first by my opponent's second argument -- that it is unclear why it was chosen that a circle has 360 degrees instead of, say 200. The answer is that it isn't entirely obvious why. My opponent references youtube video offering a variety of theories, but these are by no means the objective reason why it was chosen.
The issue at hand isn't about the arbitrary value chosen originally for degrees in a circle. My opponent clearly accepts the notion of the angle, as all of his verbal and pictorial arguments reference the degree. My opponent would no doubt agree that a square contains 360 degrees; his qualms arise when trying to decide the degrees in a circle.
A circle's amount of 'degrees' isn't based off a polygonal model for determining degrees, but rather the polygonal formula my opponent keeps referencing is based off the amount of degrees in a circle. More on that in a bit.
Sum of Interior Angles in a Polygon
My opponent has repeatedly referenced the formula for determining the sum of the interior angles of a polygon, in a variety of forms. It's simplest form can be stated as such: sum = 180 * (n - 2), where n is the number of sides in a particular polygon. My opponent makes his error in attempting to apply this formula to the circle.
As he himself admitted, a circle is not a polygon. A semicircle is also not a polygon. In fact, any shape that has a curved edge is not a polygon. A polygon is defined as a plane shape with straight sides. Attempting to apply the previously mentioned formula to a circle has disastrous results. If we consider a circle as having 0 sides, then the sum comes out to -360 degrees, a nonsense number.
Another thing we might do is assume a circle has an infinity of sides, since as you increase the number of sides in a regular polygon, the resultant shape looks more and more similar to a circle. Applying this to the interior angle sum formula, we would see that the circle contains an infinite amount of degrees. This also is a nonsense answer, in the sense that it isn't mathematically useful.
A large portion of my opponent's argument uses this incorrectly applied interior angle sum formula. Whenever we realize that some formulas simply don't apply in all cases, it becomes more clear that a circle doesn't have an infinity of degrees and neither does it have a negative amount of degrees.
Degrees in a Given Angle
Let us return to the very basics. An angle is defined as a vertex with two rays extending from it in unique directions. A polygon is, essentially, a collection of angles in which the rays of the various angles "meet up" in order to form a closed shape. However, an angle does not need to exist within a polygon for it to be measured in degrees. Consider an example my opponent referenced -- the straight angle.
(I apologize if the size is small. I am still having trouble with the images.) Here you can see a vertex with two unique areas, extending in opposite directions. This is called the straight angle because it forms a straight line. I have marked where the angle in question resides in the picture, but I haven't labeled it. First, we have to determine this angle.
Here we have laid the straight angle down onto a circle, such that the vertex of the circle and the vertex of the angle are in alignment. It is geometrically evident that the straight angle encompasses exactly 1/2 of the circle. Given that we know the amount of degrees in a circle, it is clear that the amount of degrees in the straight are exactly half that of the circle.
Now my opponent has admitted that there are 180 degree in a straight angle. In fact, what he said was that it is "one thing we can know for certain." But how can he know it for certain? Because it was taught to him this way? I imagine he was also taught that a circle has 360 degrees. Well, we *can* know for certain because we derive the amount of degrees in a given angle by how much of a circle they encompass.
My opponent (correctly) assumed that a straight angle has 180 degrees. We know this because it was arbitrarily chosen at one point that a circle has 360 degrees and a straight angle encompasses half that. If a circle had only 180 degrees, then a straight angle could only have 90 degrees.
Another example: Let's deconstruct an equilateral triangle into it's three angle by cutting each side in half, separating the shape into three identical angles, each with two rays.
By positioning each angle onto the vertex of a circle, then insuring that there are no spaces between each unique angle, we see that a triangle encompasses exactly one half of a circle, just like the straight angle! This implies that a triangle also has half the degrees a circle does (360 / 2) = 180, just like my opponent has admitted is true.
My opponent has incorrectly applied the interior angle sum formula in an attempt to deduce the amount of degrees within a circle. Furthermore, he has failed to understand that all polygonal angles are directly derived from the number of degrees in a circle. It was at one point chosen that a circle has 360 degrees and all other angle measurements have flowed from that.
The opponent implicitly acknowledges and admits this by his use of angles that conform to the 360 degree circle notion. He agrees that there are 90 degrees in a right angle and 180 degrees in a straight angle, and by application of those facts and the logic I've presented here, it is clear that a circle must have 360 degrees.
I have said in the comment section but will say here too; my opponent has missed a round, however he did tell me about this and I just want viewers to know that I am okay with that.
The below picture shows how the formula: 180 * (n - 2) provided by my opponent works. There will always be two less triangles than the number of sides. As I've proved earlier a triangle has 180 degrees, polygonas are made up of triangles, so if a polygon is made up of 3 triangles it has 3x 180 = 540 degrees. If a polygon is made up of 4 triangles it has 4x 180 = 720 degrees.
So far no formula used to calculate angles have needed a circle to have 360 degrees - The calculations are all derived from both a straight angle and a triangle having 180 degrees. I can be sure these are correct because of the below proof:
I was taught there are 360 degrees in a circle from a very early age, and that did help me to grasp what a degree measures but later in life I found it was untrue. There isn't even a need to "make up" a number of degrees in a circle because we don't write angles in more than 1 decimal place. We effectively only use a polygon with 647,640 degrees to make angle measurements. Instead of making a new unit for angle measurements it seems people have decided to just say a circle has 360 degrees without really thinking whether it does. Drawing an arch around angles encourages people to imagine the central angle is a circle, but there is no proof of this.
I showed what happens when a formula is used to calculate the degrees in a circle; it doesn't work. This shows the number 360 has been given to a circle without any mathematical knowledge, or put another way because Babylonians who couldn't even calculate Pi accurately, believed this.
My opponent says "A large portion of my opponent's argument uses this incorrectly applied interior angle sum formula", implying there is a correct way to use an interior angle sum formula. When formulas don't work, it is usually a good idea to use a tool like a ruler or protractor.
A ruler can be used to measure the circumference of circles pretty accurately despite the circumference being curved. Using a protractor, see if you get a different measurement for the curved line. I don't think you will.
If you try to measure the curved line with a protractor you should get 180 degrees, like the straight line. If any of the lines are extended the number of degrees does not change as the points making the line (or side) are parallel to each other. Even when the ends of the curved line meet the angle will still be the same.
I have created some images below to hopefully help you understand infinity and why a circle has 180 degrees, the red lines show polygons - what most people imagine, but I've included some extra lines. The blue and green distances will remain the same when new sides are added. You may think that if I keep adding new sides I won't ever create a circle or curved lines and think it'll just look similar. But an infinite sided polygon will have black lines placed EVERYWHERE. So the circle I create is made up of the centres (big red dots) of the black straight lines which there are an infinite number of.
Calculators always give the correct answer, but not necessarily to the correct question. Using a formula and multiplying 179.9... (or 180) by infinity will give the wrong answer. Most people think that a polygon with infinite sides will have lots of sides, and lots of angles. But the truth is it will have an infinite number of angles parallel to each other i.e. at 180 degrees just like a straight line.
If a circle has 180 degrees, a straight angle won't be 90 degrees because I have used a straight angle of 180 degrees to determine the angle of a circle. The exterior angles of all polygons will add up to 360 degrees e.g. 3x 120 = 360, 4x 90 = 360, 5 x 72 = 360 etc. I showed this before but with internal angles. The reason they add up to 360 degrees is because polygons are "points".
If you attempt to show the sum of the exterior angles of a circle you will just get ONE angle of 180 degrees, NOT 360 degrees like with the polygons shown, because the internal angle of a circle is 180 degrees.
I could put a straight line through any shape and say it has 360 degrees but it won't mean the shape has 360 degrees. I do agree there are 90 degrees in a right angle and 180 degrees in a straight angle, but by application of those facts I come to a different conclusion.
Suppose someone decided a circle should have 360 degrees, and they were going to work out how many degrees there are in a pentagon, how would they go about doing that step by step? Wouldn't you need to understand how to accurately draw polygons before you could divide a circle up?
First, I'd just like to thank my opponent for being so reasonable about my missed round. He could have used that against me, but chose not to. That demonstrates character and it is comforting to know that people like that use this site. Anyway, to the arguments!
I'm going to be rebutting my opponent's previous arguments in a fairly choronological order, then summarize my position and clarify my argument.
The Straight Angle Argument
My opponent first points out that the Interior Angle Sum formula [IAS] effectively uses triangles in order to determine the correct interior sum. This is, in many ways, completely correct. This leads my opponent to use the amount of degrees in a straight angle to prove that a triangle has 180 degrees, and this too is a valid proof.
This entire problem arises when we consider the amount of degrees in a straight angle. Both my opponent and I agree that a straight angle has 180 degrees. It is the reason behind *why* the straight angle has 180 degrees that our arguments differ. An important part of this discussion from here on out will be determining exactly which of our explanations hold water.
A Deconstructed Circle == A Straight Angle?
My opponent decides to make the point that if you take the circumference of a circle, snip the loop at one point, then lay the resulting curve down flat on a table, it looks suspiciously like a straight angle. And this is completely accurate. If you smooth the edge such that it is a straight line, it will certainly contain some vertex, as the line has length, so it will consist of a vertex and two straight, unique rays. This is pretty cool.
However, at the point where you do this to a circle, it is no longer a circle, it is a straight angle. The geometric measurement of an angle is essentially a way of relating different points in space. The measurement itself depends on the "static" nature of the object being measured. Many geometric shapes are, in fact, defined by how they are actually displayed on a plane.
For instance, one could imagine taking a triangle, then bowing out the edges until it forms a circle. At that point, the triangle is no longer a triangle and cannot be treated as such. This is what my opponent tries to do with his circle deconstruction. He effectively manipulates the circle until is a straight angle, measures the straight angle, then concludes that the circle must have 180 degrees. This is clearly flawed, as such manipulation would effectively give all shapes the properties of all other shapes, depending upon how you manipulated it.
For example, my opponent has agreed that a square contains 360 degrees. If you were to snip the square at a corner, then lay it flat and measure it with a protractor, you would find it also has 180 degrees. In fact, you could take any polygon and do this. If this were a mathematically acceptable way to measure things, the relevancy of 'degrees' would vanish, as it could never come to a reasonable, non-changing measure.
The Polygon With Ever Increasing Sides
Next, my opponent uses a picture that effectively demonstrates how one can increase the number of sides in a polygon to show that it slowly approaches the shape of a circle. Here is where application of IAS is useful, to see what's actually happening with the interior angles. It should be noted that my opponent erroneously assumes that this number approaches 180 degrees. Let's look:
S(n) = 180 * (n-2)
lim n--> infinity of S(n) = 180 * (n-2)
is equal to 180 * (inf - 2) == 180 * inf == inf
By applying a limit to this formula, where the number of sides approaches infinity, one can see that the interior angle sum approaches infinity and is unbounded. This is not the figure my opponent offered, that being 180 degrees. Limits are an effective way of dealing with infinity.
Now, this isn't to say that a circle has an infinite IAS. As I've mentioned before, a circle is not a polygon, and the IAS formula only applies to polygons. If you were to increase the amount of sides in a polygon, while simultaneously zooming in on a particular vertex, you would see that straight rays always jut out from the vertex. You could keeping increasing the number of sides infinitely and, if you kept zooming infinitely, you would always be able to see the individual, straight line sides. The circle is different -- you can zoom in infinitely and there will always be a measurable curve.
A Half-Way Recap
At this point, it's important to know where we stand before we continue. My opponent and I both claimed that a straight angle has 180 degree, but for different reasons. My opponent's reasoning was based of the deconstruction of a circle, which I have previouly shown to be inaccurate (and unsettling, based off its implications.) My opponent didn't make much effort to directly counter my argument that the amount of degrees in a particular angle are based off how much the angle 'sweeps' over a circle. To reiterate, a straight angle placed ontop a circle cleary 'sweeps' half the circle. This implies that the circle has twice the degree measurement that a straight angle does. Ie, 360 degrees.
I'm going to hit the rest of my opponent's arguments after this, but it's important for voters to understand that unless my opponent is able to show that the straight angle (a) is not based upon the circle and (b) has a measurement that can be uniquely derived, his argument simply can't stand.
My opponent shows that the exterior angle sum [EAS] of a polygon will always be equivalent to 360 degrees. He then argues that attempting to find the EAS of a circle results in one angle of 180 degrees. This is his flaw.
An angle, as we've previously defined, is a vertex that has two *straight* and unique rays jutting out from it. It's crucial to point out that this is where circles and polygons differ. If you pick any given vertex on a polygon, you have two straight, unique rays by which to measure the angle. If you were to arbitrarly pick a point on a circle's circumference, you would find that it *does not* fit the criteria of an angle, since it doesn't have straight rays. The left and right side of the chosen point will always be a curve. If the circle doesn't have any angles on its circumference, then it doens't have any exterior angle. Repeat: A circle doesn't have an EAS.
The measurement of a circle is unique in that it is measured from its central point, called the vertex of the circle. At this vertex, there is no angle (since there are no rays). However, we say that a circle has 360 degrees because it gives us the ability to place ontop this vertex an angle and measure the percentage of the circle that it encompasses.
Another way to think of an angle is this: The angle is the measurement of the portion of the circle it encompasses when placed ontop. And this isn't just a way to think about it -- it's the actual mechanism for giving angles measurements. The circle is the base for which all measurements flow.
My opponent asks me to demontrate step-by-step how one might use the circle to demonstrate the amount of degrees in a pentagon. He first asks how one can even accurately draw a pentagon without knowledge of angles. This is easy. It is basically like a game of connect the dots. You simply draw 5 points, then connect them with straight lines. No knowledge of angle measurement required. Let's do that.
Now that we've done this, we need to lay each of the five angle ontop a circle. It would be nice if we could do this all on one circle, but I have a feeling that the amount of degrees in a pentagon is greater than that of one circle. Let's do it!
So in the upper left corner we have our pentagon. We then break it apart into it's individual angles. We've then placed those ontop of two circles. Rather than shade the part they encompass, I've shaded the part that they don't. Finally, we can take those two shaded reasons and place them atop another circle.
In the upper right, we see that the angles plus the unencompassed parts equals two circles. When we look at the shaded regions alone, we see they encompass exactly half a circle. By subtracting the shaded part from the angle part, we get 2 - 0.5 = 1.5 circles. If we assume the circle has 360 degrees, this would imply that the pentagon has 1.5 * (360 degrees), or 540 degrees.
By using the IAS formula, we can show this is true.
S(5) = 180 * (5-2) = 180 * 3 = 540 degrees.
This identical measurement would only be possible if the circle had 360 degrees.
I've demonsrated that the circle has 360 degrees by showing that the implications of such is the only thing consistent with our current understanding of how angles work in polygons. My opponent has used various 'proofs' that the circle has 180 degrees, but I've shown all these to be faulty in some way or another.
Who knows why it was decided that the circle has 360 degrees? No one knows for sure. What we do know, however, is that we can only measure angles based upon the amount of a circle they encompass. My opponent has accepted these implications by admitting that a triangle has 180 degrees, that a straight angle has 180 degrees, that a square has 360 degrees, etc. By accepting the degree as it applies to polygons, he has no choice but to accept that it is directly based off the measurement of a circle. Without the circle, we would have no way of know that a straight angle did indeed have 180 degrees, or any other measure.
Sorry for this particularly long round. I hope it was worth the read.
Imagining a circle or any other shape as a string that can be layed out flat is not really a good way to picture my argument. If you look at the two lines agian:
The curved and straight line do appear exactly the same, but the curved line is a very small part of the biggest circle you can imagine. Only by extending the lines will it be clear which line is curved and which is straight. If you could place a protractor on the curved line and read the angle it will be 180 degrees. Enlaraging any shape does not change the angles.
Infinity is a difficult thing to fatham, and I only ask voters to try to understand my 4 diagrams. A polygon with infinite sides will only have one side, my opponent disagrees and says "if you kept zooming infinitely, you would always be able to see the individual, straight line sides. The circle is different -- you can zoom in infinitely and there will always be a measurable curve". So according to my opponent a circle and an infinite sided polygon are two different things which just appear similar from a distance.
There will always be a polygon with more sides. If you stop at the largest finite polygon you can think of and draw it in the same way I've shown with black lines extending from the polygon, the outside will seem jagged, even if you keep adding more sides! An infinite sided polygon though won't have any "missing" bits of black due to a jagged surface, because there are an infinite number of sides filling EVERY space at the distance of the blue arrow minus the green arrow distance.
My opponent uses an equation to show that an infinte sided polygon will have an angle of infinity, below is that equation
"180 * (inf - 2) == 180 * inf == inf"
This equation does work for finite sided polygons e.g. if you have a 20 million sided polygon the sum of the angles will be 180 x (20 million - 2) = 3599999640. Each angle will be 3599999640 / 20 million = 179.999982. You can also work this out the other way by using the equation 180 - (360/20 million) = 179.999982 degrees, and multiplying this answer by the number of sides to get the sum of the angles. I could keep thinking of bigger sided polygons and the sum of angles will keep getting larger and larger however the polygon that I think of will always have finite sides and NOT infinite sides.
An infinite sided polygon has one side, and an angle of 180 degrees. Multiplying 180 by infinity is a crazy thing to do, it would be like adding every number you can see in the below picture to find the number of degrees in a rectangle.
I'm not quite sure why my opponent brings up bowing out edges of polygons until they form a circle, as this is certainly not something I have done and does not support his argument. I'm also not sure where he said anything about the amount of degrees in a particular angle being based off how much the angle 'sweeps' over a circle. He does reiterate saying a straight angle placed ontop a circle cleary 'sweeps' half the circle but have no idea what he means.
According to my opponent a circle has no exterior angle. Every shape including a circle can be put on top of a straight line whereby the angle above the line is 180 and the angle below is 180 - see last pic in round 2. I've made another picture below to show exterior angles. A pentagon has 5x 72, a hexagon has 6x 60, a 20 million sided shape has 20 million x 0.000018 etc. The more sides that are added the lower each exterior angle becomes but the sum of them will still equal 360. However when they equal nothing due to having infinite sides nothing the angle outside of the circle is just 180 meaning the angle inside a circle is 180 degrees.
Apparently I must show that the straight angle is:
a) not based upon the circle
b) has a measurement that can be uniquely derived
A turn is 360 degrees, but you can make a turn in an infinite number of ways by walking along the sides of different polygons, not just round in a circle. I have never needed to know a circle has 360 degrees to calculate angles, also straight distances have always been easier to measure - why the Babylonians didn't understand circles too well, so it makes sense that angle measurements were derived from triangles having the same number of degrees as a straight angle. My opponent attempted to show how to work out the number of degrees in a pentagon when the angle of a circle is chosen to then can calculate other angles. You will see that before he can divide a circle or put angles on a circle he does need to understand how to accurately draw triangles with specific angles, meaning angles are not based from a circle.
I could not have hoped for a better opponent! I'd like to thank you for going out of your way to challenge me and taking my arguement seriously
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