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So.. Of course the area of a circle is πr2.. 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 πr2.

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...

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(Deleted comment)
Really? I didn't think I explained it all that well...

The 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)

If there is such a thing as telepathy, I'm fairly certain you've done it. Last night on my half hour drive to my friend's house, nearly the entire conversation I had with myself was about the geometry of circles :P

I am pleased to inform you that this thought entered my head at about 4pm your time, which probably pre-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)

Ah yes. The geometry entered my head around 5pm... Well done ;) Not that I mind so much, we're generally on the same page about most things. So really, you'd just be reinforcing my incredibly annoying "I am right" attitude :P

If this is what you consider inconsequential, I am in a big fat lot of trouble.

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