*course*the area of a circle is πr

^{2}.. I mean, it's obvious, though I don't think that anybody ever covered it explicitly at school. Not on a day when I was there, anyway..

So, taken to the extreme, a circle is just a set of triangles radiating from a central point. I mean, when you subdivide a circle into infinite sectors, then the outer edges are all locally straight, after all..

And because the division is infinite, and you have that local straightness, the angles in each triangle approach {90,90,0}, which means that the

*area*of each is equal to rx/2 where x is the length of the outer edge (which is basically zero in an individual case but which can be summed over the entire circle).

So, to sum that lot over the whole circle, what you'll ultimately end up with is a

*total*edge length that is equal to the circumference of the circle, which is πd, or 2πr. And you can factor the r/2 out, because it's common to all the summed items, so what you end up with is r/2 . Sum(x), where Sum(x) = 2πr, ergo the area of a circle is πr

^{2}.

Now I get that this is

*really*trivial, but I think it's possibly one of those things that was

*so*trivial that it never got explained, at least not in a lesson that

*I*was present for. Though I was kind of bad at that sort of thing.

In any case, this thought popped into my head when I was trying to get to sleep last night. Funny when these things will strike you...

(Deleted comment)unknownjThe version in my head has all kinds of pretty formulae that all work their way to the bottom, I just lack the ability (or inclination) to type it all out that way :o)

ashleybrooke7unknownjprobablypre-dates your thought, and means that I'm broadcasting, rather than receiving.Which is, of course, exactly how I'd want it. Who cares about reading people's minds? I want to write to them.. :o)

ashleybrooke7starthieving