Why is it πr² for the area of a circle?

It is well understood that π is simply the ratio between the circumference and the diameter of a circle.  Well where did the formula for the area of a circle come around.  Here’s a proof for this which I found highly interesting.  Ok so first you take a circle and make an even number of alternating sections of two colors.  Take these sections out and overlap the ones of different colors to make a sort of rectangular form.

figure27-2.pngAs seen in the image, in the rectangular shape of the sections, the length is half the circumference, as it is the sum of the arcs of one color, so is πr, and the height is simply r.  Now this shape is not exactly a rectangle, so we better keep a very large number of sections, or rather an infinite number of sections so that the length of each individual section in its essence the same as the chord which would cut across it, and so the shape is a rectangle.  So what is the area of the rectangle?  πr x r = πr².  And this is the intriguing visual proof for the area of the circle.  Cool, huh?


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