The area of a circleWatch
why does the area of a circle = pi * r^2
This page explains the non-calculus way i think, although it's not a totally acurate method, it's accepted as being a reasonable demonstration of how the area equation works.
However, you can think about it more informally...all the formula says is that the area of a circle is proportional to its radius*radius (or diameter*diameter)...since the circle is a symmetrical shape, this should be fairly obvious. A square's area is proportional to length*length as well.
using calculus, or not?
go for it...i want any type of proof
Now we define there to be 2pi radians in a circle (because that allows us to do radius*angle = arclength), so if we integrate our tiny triangles between 0 and 2pi:
Int(0.5r^2) between = 2pi and 0 =
(since r is a constant) [0.5r^2x] between 2pi and 0 =
[0.5r^2 * 2pi] - [0.5r^2 * 0] = 2*0.5pi*r^2 = pi*r^2 as required