The Instigator
mostlogical
Con (against)
Losing
0 Points
The Contender
BoggyB
Pro (for)
Winning
7 Points

Are there 360 degrees in a circle?

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Post Voting Period
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after 1 vote the winner is...
BoggyB
Voting Style: Open Point System: Select Winner
Started: 4/7/2015 Category: Education
Updated: 1 year ago Status: Post Voting Period
Viewed: 726 times Debate No: 73040
Debate Rounds (3)
Comments (7)
Votes (1)

 

mostlogical

Con

I will give my argument as to why there isn't 360 degrees in a circle, whilst Pro will argue why there are 360 degrees in a circle

Let it begin
BoggyB

Pro

Intro:

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.[1].

"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 [2].

Since Con is the one making the substantial claim, BoP will be on him.

Citations:

[1]. http://dictionary.reference.com...

[2]. http://i.word.com...
Debate Round No. 1
mostlogical

Con

Good day. Thank you for providing definitions, and I accept the burden of proof.

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 [1].

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

sources:

[1] http://en.wikipedia.org...
BoggyB

Pro

Intro:

First I will give my argument as to why a circle is 360 degrees, and then I will rebuttal Con's arguments.

Constructive 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. [1]. 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.

90 degrees:

We can take a graph [2], 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. [3]. 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. [4]. 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. [6].

180 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 [5]. That means 180 degrees is on either side of the diameter. 2 x 180 is 360.

Rebuttals:

"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. [7]. 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)" [8].

So the method Con applied to show circle isn't 360 degrees doesn't hold any water, so to speak.

Conclusion:

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.

Citations:

[1]. http://www.mathsisfun.com...
[2]. http://etc.usf.edu...
[3]. https://www.mathsisfun.com...
[4]. https://whites-geometry-wiki.wikispaces.com...
[5]. https://www.mathsisfun.com...
[6]. http://www.mathopenref.com...
[7]. http://www.math-play.com...
[8]. https://www.mathsisfun.com...
Debate Round No. 2
mostlogical

Con

Definitions help to determine where someone stands in a debate, I do not believe it is possible to win by definition, if for example the moon was defined as a ball of cheese it doesn't mean it is. However I do understand why you think that if we define a measure then it is that.

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 [1] 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 [2]

[2] The sum of internal angles = (n-2) x 180 where n = number of sides.
[2] 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

sources:

[1] http://www.mathopenref.com...
[2] http://www.mathsisfun.com...
BoggyB

Pro

Rebuttals:

"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 [2]"

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)." [1]. 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. [2].

Conclusion:

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.

Citations:

[1]. https://www.mathsisfun.com...
[2]. http://www.math-play.com...
Debate Round No. 3
7 comments have been posted on this debate. Showing 1 through 7 records.
Posted by mostlogical 1 year ago
mostlogical
I hope you agree with me here, if someone decided a long time ago the sun was 100 miles away for whatever reason despite being able to measure things in miles on Earth, and everyone accepted this as fact, but many generations later someone accurately approximated the distance of the sun, people should accept that whoever said the sun was 100 miles away was wrong.

True, someone created the concept of degrees just like with any measurement, and used the number of degrees in a line (not a circle) to measure the angles inside polygons. When someone says a circle has 360 degrees they are also saying that a square has the same number of degrees, and it has twice the number of degrees as a triangle etc. Basically they are thinking 'it is not possible to know, so let's give it a value' just like with the example I have shown above about the sun.

You say that a curved line can't be given an angle, yet a circumference of a circle can be measured exactly despite that line being curved, of course no-one can write it down exactly because it would go on forever but that's why we use approximations for Pi such as 22/7. With angles, you can get better approximations; I've shown the method to do that - by putting a polygon inside a circle with more and more sides. A straight line has one side, and a curved line (or circle) has one side too, which is why the number of degrees in a circle are almost identical to those in a triangle, not a square.

I'm sure there are formulas which can show how much a line "appears" to have curved, but this is not the same thing as finding the number of degrees inside a shape.
Posted by BoggyB 1 year ago
BoggyB
I argue that a circle should be accepted for having 360 degrees because it was created. Someone created the concept of "degrees." It wasn't discovered like you seem to think. If I took a right angle and divided that angle into 9 parts, and called each part a "chunk," I would've created a new measurement. There would be 4 x 9 "chunks" in a circle=36. Some decided a circle should be divided into 360 different parts, and each part would be called a "degree."

I finally understand what you are trying to convey with your argument. Unfortunately it is incorrect. Since the first 15cm line is straight, we can say it is 180 degrees. The second line can't be attributed an angle because it is a curved line. Angles can only be measured between two straight lines connected at a vertex. A curved line has a different formula for determining its curve than a straight line. if we took both your 15cm lines and connected them at a vertex, we wouldn't be able attribute an angle in degrees to that angle, because both of the lines need to be straight. Likewise, the curved line can't be given a degree like you say, because for it be curved, means the level of degree wouldn't be constant like you claim. It was a good attempt at trying to think outside the box, but it defies the laws of math and isn't true.
Posted by mostlogical 1 year ago
mostlogical
You argue a circle should be accepted as having 360 degrees because someone decided, but if a long time ago someone decided an octagon should have 400 degrees rather than 1080 because it seemed special in some way, would that mean an octagon has 400 degrees? Or an anomaly?

If you imagine a straight 15cm line beside another line 15 cm long that's curved but which appears straight, and use your protractor you will find the second line has slightly less than 180 degrees. If that second line was extended to a metre so you can see it's curvature, the angle of that line has not changed, it has remained exactly the same. I've just calculated that angle and demonstrated how the angle can't become a whole number i.e. 180. When the ends of the line come close to meeting it still has slightly less than 180 degrees, and when they meet it will not change this fact.
Posted by BoggyB 1 year ago
BoggyB
In reality, a polygon with infinite sides might look like a circle with our eyes, but we aren't talking about a situation like that. We were debating a mathematical principle, which is a circle. I'm arithmetic, a circle has ONE, CURVED side. Notice "one," and "curved." A polygon, must have at least THREE sides that are all STRAIGHT. No matter how close to a circle a polygon might be, it isn't, because it is made up of multiple STRAGHT sides, while a circle is one CURVED side. There is nothing else to say. Your logic has been mixed up, and is leaning towards rejection and denial of mathematical facts, rather than intuitive thinking.
Posted by mostlogical 1 year ago
mostlogical
I have read your arguments, and I've explained that there is no difference between a circle and a polygon with infinite sides, so they can be treated in the same way. If they were different then the rationality and value of Pi would be unknown.
Posted by BoggyB 1 year ago
BoggyB
The reason your logic doesn't work is because a circle isn't a polygon. Did you even read my debate? I stated it numerous times throughout. Go back and see all the times I explained it and the sources which also show it is not a polygon. A circle isn't infinite sides it is one side, a curved line. A polygon has at least three sides (a circle has one) and all sides of a polygon are straight (a circle has one curved line). A circle is not a polygon, thus your method doesn't work.
Posted by mostlogical 1 year ago
mostlogical
My logic is based on a triangle having 180 degrees, this means one corner of a square will have 90 degrees i.e. twice as less. If you imagine the lines from a square extending to give you this shape # you can see a line has 180 degrees like a triangle because 90 + 90 = 180. Changing the number of degrees in a triangle does not change the fact a triangle has the same number of degrees as a line.

You can apply this logic to calculate the angles for any sided polygon (they can all be divided into triangles) including one with infinite sides i.e. a circle, and you will find a circle does not have the same number of degrees as a square.
1 votes has been placed for this debate.
Vote Placed by Kozu 1 year ago
Kozu
mostlogicalBoggyB
Who won the debate:-Vote Checkmark
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.