Is pi really 3.14159...?

[Gontlemang]

In this paper, I think I may have disproven Archimedes's geometric proof of pi. His method involves bounding a circle between 2 polygons. He assumed that as the number of sides of the polygons increase, the circumference remains between the perimeters. It's true that the circumference is always larger than the perimeter of the inner polygon. This can be proven. However, it has never been proven by anyone that the circumference always remains lower than the perimeter of the outer shape. We just assume that it is. So I set out to test this theory and it turned out to be wrong. Past an n number of sides pi goes above both bounds. My proof involves bounding a unit circle between two 360 sided polygons. I then compare one side of the outer polygon with one corresponding arc. I approximate the arc length with chords and that approximation is larger than the side length of the polygon, disproving the pi value of 3.14159...

I would like to hear your thoughts on these findings and if I am wrong I am open to hearing your critique. Thank you.