Are there 360 degrees in a circle?
Debate Rounds (3)
Let it begin
I'll define a few words to start:
"Circle" - : a closed plane curve consisting of all points at a given distance from a point within it called the center..
"Degree" - :a unit of measure for angles equal to an angle with its vertex at the center of a circle and its sides cutting off 1R60;360 of the circumference; also :a unit of measure for arcs of a circle equal to the amount of arc that subtends a central angle of one degree .
Since Con is the one making the substantial claim, BoP will be on him.
Many people including myself have been taught there are 360 degrees in a circle. To understand why this is incorrect I think we have to look at the history; it's not clear why 360 degrees was chosen, it could be because there are about 360 days in a year, or due to the Babylonians .
So basically, we have accepted there are 360 degrees in a circle on faith!
I will now quickly show you how to calculate the internal angles of polygons
let's take a square which as you know has 4 sides,
360/4 = 90 degrees
There are 180 degrees in a triangle, so the other angles add up to 180 - 90 = 90
There are 4 sides, so a square has a total of (4 x 90) degrees = 360 degrees
If we have a 16 sides shape now
360/16 = 22.5 degrees
There are 180 degrees in a triangle, so the other angles add up to 180 - 22.5 = 157.5
It will therefore have total of (16 x 157.5) = 2520 degrees
If we have a million sided shape now
360/1 million = 0.00036
There are 180 degrees in a triangle, so the other angles add up to 180 - 0.00036 = 179.99964
It will therefore have (1 million x 179.99964) = 179999640 degrees
As you can see the more sides there are in a polygon the closer the angle "OF ONE SIDE" becomes to 180. A circle only has one side! In fact I can place any numbered side polygon inside a circle and the angle will never reach 180, because only a straight line has 180 degrees. The internal angle of a circle is therefore 179.99... degrees
First I will give my argument as to why a circle is 360 degrees, and then I will rebuttal Con's arguments.
A degree is a form of measurement where 1 full turn is 360 degrees. I degree is 1/360 of a full turn, or circle. . So, technically by definition I have won. Although, this is sufficient enough for us to realize there are 360 degrees in a complete circle, I'll go a little bit more in depth.A right angle is 90 degrees. A straight line is 180 degrees. A circle is 360 degrees. We can come to the conclusion of 360 degrees through either a 90 degree explanation or 180.
We can take a graph , and plot a point at (0,0). X=0 on the X-axis, and Y=0 on the Y-axis. This will be the center of the circle. Each quadrant of the graph is 90 degrees. Each corner is 90 degrees. A circles radius is the same length from the center out all the way around the whole circle. . Let's say the radius for our circle is 2. The center is (0,0) with a radius of 2. If we move two clicks up and down, left and right on the y, and x axis, it would appear like this. . We see this is a circle the envelops all four of the right angles. 4 x 90 = 360. A circle contains 360 degrees. The circle doesn't have to be on the origin (0,0) for the circle to be 360 degrees, this was just an easier example to illustrate that a circle will have 4, 90 degree angles from the center.
A second 90 degree example is using tangents. A tangent is a straight line that touches the circles edge at one point. If we drew the radius from the center out to the point where the tangent touches the circle, it will form a right angle (90 degrees). We multiply this by 4 to find again 360 degrees. .
If a line is 180 degrees, we can make a circle around that line. The diameter goes through the center of the circle. It is a straight line . That means 180 degrees is on either side of the diameter. 2 x 180 is 360.
"So basically, we have accepted there are 360 degrees in a circle on faith!"
No we haven't. It's not faith, it's a simple concept. Degrees of an angle is just a term of measurement. Years ago 360 was chosen as the number, because like you said it reflected the calendar. Although it doesn't really matter why specifically 360 was chosen, all that matters is that it was. It was chosen that the unit of measurement would be degrees, and their would 360 individual degrees in one full turn of a circle. It wasn't faith, it was just created that way. Mathematicians didn't struggle years to discover this figure of 360, they created it precisely with 360 degrees in a full turn."I will now quickly show you how to calculate the internal angles of polygons."Unfortunately, this method you used isn't mathematically sound. What Con has done is show three examples of polygons and calculated their internal angle. He used a square: 4 sides, 4 right angles, and sixteen sided polygon, and a million sided polygon. For each example he showed the internal angle to by just under 180 degrees. So he concluded that the more sides, the closer to 180 degrees the internal angle gets.
The catch is that a circle is not a polygon. . A polygon is: "A plane shape (two-dimensional) with straight sides.
Examples: triangles, rectangles and pentagons.
(Note: a circle is not a polygon because it has a curved side)" .
So the method Con applied to show circle isn't 360 degrees doesn't hold any water, so to speak.
I have answered the resolution to its fullest extent, by showing a circle does have 360 degrees, and have also shown how Con failed to show why a circle doesn't have 360 degrees.
A number must be "chosen" for measuring and comparing angles. It could be any but it makes more sense to work with straight lines rather than curved lines. A straight line is defined as being an angle whose measure is exactly 180 degrees  By using very basic trigonometry we can neatly divide a circle into 24 slices (15 degrees each) so it doesn't have to be divided into 4 slices (90 degrees each).
The degrees in a shape should tell you something about that shape. If we just accept a circle has 360 degrees, how many degrees would there be in a different shape like square? It certainly wouldn't be a nice number to work with, nor would it for any polygon.
"90 degree proof"
I could place any polygon inside a circle, not just a square, and say the circle envelops each of the angles, and even place the circle inside of a polygon to show tangents.
"180 degree proof"
I could place a line in the middle of any polygon such as a hexagon or pentagon and say there is 180 degrees on either side and wrongly conclude those shapes have 180 degrees using your method.
A polygon with an infinite number of straight sides perfectly represents a circle with a curved line. If this was not true the rationality of Pi would be unknown, but it is known.
The method I used to calculate internal angles of polygons is mathematically sound, you can check it is correct in many ways. E.g. a person can select a corner of a polygon and draw lines to each corner, creating triangles inside, each triangle has 180 degrees so if you count the triangles inside and multiply this figure with 180 (degrees in a triangle) you will arrive with the total internal angle of that shape and then if you like you can divide the total internal angle by the number of sides to give you the angle for each corner 
 The sum of internal angles = (n-2) x 180 where n = number of sides.
 Each angle = (n-2) x 180/ n
I only showed a few examples, but I'm sure you can see a pattern.
If we have a 5 million sided shape now
360/5 million = 0.000072
There are 180 degrees in a triangle, so the other angles add up to 180 - 0.000072 = 179.999928
It will therefore have (5 million x 179.999928) = 899999640 degrees
A circle only has "one corner" if you like, and if you try to calculate this for eternity you will get 179.99...
I will remind voters that this debate is about whether there is 360 degrees in a circle. I do not need to prove the exact value, only that there is not 360 degrees which I believe I have done.
Any comments are welcome
Thanks for having this debate
"By using very basic trigonometry we can neatly divide a circle into 24 slices (15 degrees each) so it doesn't have to be divided into 4 slices (90 degrees each)."
It appears as if Con has conceded the debate here by showing there are 360 degrees in a circle. He says we can divide a circle into 24 even slices at 15 degrees each. 24 x 15 = 360.
"The degrees in a shape should tell you something about that shape. If we just accept a circle has 360 degrees, how many degrees would there be in a different shape like square? It certainly wouldn't be a nice number to work with, nor would it for any polygon."
We are not just "accepting" a circle has 360 degrees. We know for a fact there are 360 degrees in a circle, because it was decided, not discovered. Someone decided that a circle, must be divided up into measurable distances and angles to be measured. They chose a name for the measurement (degree), and a number which to divide that circle up into. They chose 360, so that if we take 360 degrees from one vertex, we get a circle. Also, you cannot compare the circle to the square in this situation, because a square is a polygon, and a circle is not.
"I could place a line in the middle of any polygon such as a hexagon or pentagon and say there is 180 degrees on either side and wrongly conclude those shapes have 180 degrees using your method."
Actually you wouldn't be able to apply my method to "any polygon." My method applied to a circle (not a polygon) and you are trying to force upon a polygon.
"The method I used to calculate internal angles of polygons is mathematically sound, you can check it is correct in many ways. E.g. a person can select a corner of a polygon and draw lines to each corner, creating triangles inside, each triangle has 180 degrees so if you count the triangles inside and multiply this figure with 180 (degrees in a triangle) you will arrive with the total internal angle of that shape and then if you like you can divide the total internal angle by the number of sides to give you the angle for each corner "
You are continuing to show your incompetence. Your method is not sound. Your whole entire argument hinges on the concept of interior angles. I will point the debate itself was not about interior angles. Not only that, but you are trying to show the interior angle of a circle. This simply doesn't work. You have been continuing to take a method for calculating the interior angle of a polygon, and apply it to a circle (not a polygon). I already explained it last round, but I'll do it again. A polygon is "A plane shape (two-dimensional) with straight sides. Examples: triangles, rectangles and pentagons. (Note: a circle is not a polygon because it has a curved side)." . A polygon has straight lines. Notice the words straight and the plural lines. A circle has a curved line, while a polygon has a straight line, and the plural lines means more than one, while a circle only has one line. .
Con didn't stay on track to the resolution, by stubbornly trying to prove a circles interior angle, which wasn't the topic of the debate. Not only that, but I proved that a circle is not a polygon, so his theorem didn't even apply to circles.
Conversely, I stayed on track, and showed how there are in fact 360 degrees in a circle.
Thank you, please vote Pro.
1 votes has been placed for this debate.
Vote Placed by Kozu 1 year ago
|Who won the debate:||-|
Reasons for voting decision: Con does a great job of working around logic. Unfortunately he doesn't offer any other method to measure degrees other than what the current status quo currently orders. He can question the definition of a circle or how many degree's it has, but he has no foundation to apply his logic to other shapes or mathematical theorems. He's simply speculative.
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