# The area of a circle

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#1
why does the area of a circle = pi * r^2
0
15 years ago
#2
(Original post by GayBoy)
why does the area of a circle = pi * r^2
using calculus, or not?
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.
0
15 years ago
#3
A formal proof can be done using integration.

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.
0
#4
(Original post by Louise_1988)
using calculus, or not?
go for it...i want any type of proof
0
15 years ago
#5
(Original post by GayBoy)
go for it...i want any type of proof
posted above!! (i'm not a pro-maths dude, but my teached did explain this one to us. Somebody who knows more on the subject may be able to explain better!)
0
15 years ago
#6
imagine you draw a tiny tiny tiny segment of a circle. It's effectively a triangle. A triangle's area is .5(base*height). Now the base is easy, that's just r. But the height? Well if we call the tiny angle dx and it is in radians, then the height of the tiny segment is effecitvely it's arclength, which =rdx. So The triangle area = 0.5r^2dx

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
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