There are 360 degrees in a circle.

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 circleb) has a measurement that can be uniquely derivedA 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!