There are 360 degrees in a circle.

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 SidesNext, 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 == infBy 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 RecapAt 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. Exterior AnglesMy 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. The PentagonMy 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. ConclusionI'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.