I had this idea of calculating the area of a circle as the sum of circumferences where the radius is being shortened by an infinitesimal amount until it reaches 0. I was told that this was impossible, because an area can only be represented as a sum of smaller areas, but not lines. I tried to make a program that calculates the area of a circle by using this method, here's the algorithm:
This works, I've tested it with numerous examples. The bigger the n, the greater the precision.
My 2 questions are:
- Why is this working if an area cannot be represented as a sum of lines?
- Why must I divide the result given by this method by the reciprocal value of the value I shorten the radius with? As I said, the greater the n, the greater the precision. Thus this will be the exact area when n approaches infinity, but wouldn't I be dividing by infinity later? How does this give me an exact result?