[EricH]

Originally Posted by Sovietmagician

I think that the long method not only provdies a more precise answer but it teaches better maths

I couldn't understand it, and I'm a little beyond even AP Calc or whatever. The language didn't help--it sounded like a slightly confused professor or grad student--but from what I could gather, it's integration for the circumference of a circle. That's nice for a proof, but we aren't looking for a proof (ugh...geometry), we're looking for the circumference of a circle.

I'm not even sure where the triangle comes in...

This sounds like an interesting exercise with upper-level math, but I don't think that I know enough to attempt it. I probably also won't for a long time. (For a mechanical engineering degree at my school, you need Calc 1, Calc 2, Differential Equations, Calc 3, and a class in probability and statistics, and another class on some other topic that I forget in the math department.)