Let's assume you have a circle. Take the circle, and draw a straight line through it. It does not have to be in the center. It can be near the top, if you want. Make sure you mark the points at each end. Good. Now take one of those points, and make a line that is perpendicular, that is 90 degrees opposite of that line, and dot both ends (one will already be dotted due to it being based off of the same point as the original line). Now you should have two lines that make a right angle. THIS IS VERY IMPORTANT TO UNDERSTAND THE FOLLOWING PROCESS!!! Let's name the lines 'a', 'b'.
If we were to make a midpoint at both lines, it would divide the length in half. Take both midpoints and connect them. The formula for this would be the square root of ( a ^ 2 + b ^ 2), divided by two. Alternatively it can be the square root of (a ^ 2 / 4 + b ^ 2 / 4) or as the square root of [(a ^ 2 + b ^ 2) / 4]. The result we will call y.
Now lets go back to the original lines a and b. Use the pythagorian theorem to figure out the length of the hypotenuse if the ends of the lines were connected. Let's call this value z.
z times pi is equal to the perimeter of the circle.
y times pi times 2 = z, so it is also equal to the perimeter of the circle.
If we took half of z, squared it, and multiplied it to pi, we would get the area of the circle. This means that squaring y and multiplying it by pi will also equal the area of a circle.
2y = z.
y is the radius and z is the diameter.
So here is my theory.
Taking two perpendicular lines that meet at any point on the circle's perimeter and dividing both lengths by half, adding each squared length and dividing the sum by 4, and square rooting the result will be equal and multiplying by 2 pi will equal to squaring both sides, adding them, square rooting the sum, and multiplying the result by pi. Dividing the result, prior to multiplying by pi, by two, squaring it and multiplying by pi will be equal to taking the first formula after dividing by four, and squaring that and multiplying by pi.
After discovering this, I think it is obvious....
...I was bored in accounting today.