ID:109369
 
<!--

So, as many of you would've been taught in middle school or high school, if you have a circle of radius r, then its circumference is 2×π×r and its area is π×r². Likely, you were simply told to memorize this, and never given any sort of explanation about how or why it was true. There is a very, very simple proof using basic calculus.

First, as above, assume we have a circle of radius r. Then we have an infinitely large set of circles of smaller size, between 0 and r. To express this in mathematical notation, the set of [0, r]. If we have a function Circ(r) = 2×π×r (basically, the circumference function), and add the result of this function for every number in this set, then we will get the area of the circle of radius r. But how are we going to do this? Integration.

First, we'll start off with the following integral:

0r 2×π×t dt

This means that we take the integral of the function 2×π×t with respect to t, and we do so from 0 to r. As 2π is a constant, it can be moved outside the integral:

2π × ∫0rt dt

Now we find the antiderivative of t with respect to t. (If you know what derivatives are, but not integrals, then know this. An antiderivative of a function f is a function F such that F' = f):

t dt = t² / 2

The indefinite integral of t is t-squared divided by two. Now we plug the indefinite integral back in, and we get:

2π × [t² / 2]0r
= 2π × [r² / 2 - 0² / 2]
= 2π × [r² / 2 - 0]
= 2π × [r² / 2]
= (2×π×r²) / 2

And, ultimately, the 2's cancel. This leaves us with:

π×r²

Because our initial integral was exactly equal to the area of a circle of radius r, and because our integral simplifies to this, then therefore the area of a circle of radius r is exactly equal to π×r². Q.E.D.

-->

NOPE.
I feel Smarter :D!
Just learned that recently =P
Basic stuff.
.>'
Naokohiro wrote:
Basic stuff.
.>'

Very, very, very, very basic calculus, yes.
My eyes glazed over when reading this.
I never really had it explained why circumference = 2 x Pi x Radius.
pi is the ratio of any circle's circumference to its diameter, and the diameter if a circle is 2R, so multiply that by pi...
magic all up in this bitch, how did they work out it was Pi, I wonder?
Stephen: Yeah, that's what I think about. This integral was an easy find, but I want to find the origins of pi.

[Edit: But I really don't care that much. I'm not a math major.]
What I told Stephen over MSN was, take some tape, measure the circumference of something round, then measure the diameter. pi is defined as the ratio of the circumference to the diameter. So simply calculate c/d. The more accurate you are the closer the result will be.

Also google "Buffon pi".
Which makes sense, because Circ = pi(d)
Yeah, pi is simply a ratio of two numbers, as the ratio between the radius (and diameter) and the circumference of a circle is constant. You don't necessarily have to know the value of pi, just that it's constant. The math can be worked out purely via the constant without the value.