My name is Stephen I am 10th grade student. I would appreciate if you could please tell me if what I discovered here is something or my ignorance? I noticed that a circle with r radii has the folling characteristic.

r = [2 * ( pi * r² / pi * 2r)]

The equation states that the ratio of a circles area over its circumfrence = 1/2 that of the circles radii. It works every time. Did you know this ? Is it some kind of therom and can it be used for any thing? I thought this was intresting and would appreciate any input you may have.
Thank you.

Hi Stephen,

NEAT! What you have done is what kids are supposed to do -- they should discover results for themselves and not wait around for the teacher to feed them the formulas. What you have discovered was perhaps known to Euclid, but the details were first worked out by Archimedes around 200 BC. Everybody who understands geometry realizes that the circumference of a circle grows as its radius, while the area grows as the square of the radius. It takes special insight to realize that the growth factor is the same in both cases, namely the number we denote by pi. It turns out that your observation is exactly what is needed for the use of trigonometric functions in calculus.

Cheers
Chris