There are 360 degrees in a circle.

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 PolygonMy 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 AngleLet 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. Closing RemarksMy 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.