Computing the Area of a Circle
My father once teached me this technique a long time ago but i don't know where he saw it; probably in his lost school book.
Start again from the begining but cut the circle in eight equal parts, for example. Rearrange like in the next figure. Each pie is like the ones before, the only diference is that the round border has a lenght that equals the perimeter divided by eight:We can do this cutting thinner pies, but an even number of it in order to obtain as many equal oriented pies has those in opposite directions.So in the limit of very (infinite) thin pies we get a figure that is a rectangle with two paralel borders who's lenght equals the radius of the original circle and each of the other paralel borders has the lenght of half perimeter. It's easy to guess that the area of this rectangle equals the area of the original circle and so
I ask you to recall what i said about integration in a post. The difference here is that there are no remaining parts left during the process because we are using pies instead of squares or rectangles. In the limit of very small pies, each pie will look more like a very thin rectangle. That's what easies the calculation: we know how to calculate the area of a rectangle and thats enough.