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October 19

Area of an ellipse

After a long search, I just found this website matematicasvisuales, and I learned from this a very intuitive way to prove the $\pi ab$ ― as intuitive as the method to prove the area of a circle.

I must ask, how come everyone associate this topic with high advanced mathematic, and the only way you can find everywhere for proving this includes Calculus, Integration etc.? The greeks had proved the circle's area formula the old "intuitive" fashion way, and that's how we mostly learn it.

In the Area its written like this:

The area formula $\pi ab$ is intuitive: start with a circle of radius $b$ (so its area is $\pi b^{2}$ ) and stretch it by a factor ${\frac {a}{b}}$ to make an ellipse. This scales the area by the same factor: $\pi b^{2}\left({\frac {a}{b}}\right)=\pi ab$ .

That is so not-intuitive; just like $2\pi R$ doesn't claim a word about the area being $\pi R^{2}$ . Sorry. יהודה שמחה ולדמן (talk) 19:25, 19 October 2016 (UTC)[reply]

@יהודה שמחה ולדמ: The fact is, this is not a rigorous proof, unless you go into advanced mathematics anyways: you have to be careful about what it means to "stretch" or "shrink" and why area is multiplied by the same factor. The circle's area formula has an "intuitive" proof but intuition is not enough; to make the Greeks' proof sound one must resort to mathematical analysis techniques anyway. There's no avoiding it.--Jasper Deng (talk) 19:33, 19 October 2016 (UTC)[reply]

That's not what I said; the wiki-page of Area said "stretch"! I didn't say the proof is rigorous either; and even if it isn't, why aren't the circle and ellipse treated the same way and not been taught together in school, or at least be taught together only through Calculus?

And by the way, I used a little Analytic geometry together with the circle and ellipse equations, and proved that the areas of the Trapezoid/Triangle cross sections relate to one other respectively as

{\frac {a}{b}}

or

{\frac {b}{a}}

. יהודה שמחה ולדמן (talk) 20:12, 19 October 2016 (UTC)[reply]

@יהודה שמחה ולדמ: But "stretch" and "shrink" are equally intuitive then, and the two "proofs" by "stretching" (as in the article) and "shrinking" (as on the page you linked) are equivalent. Both formulae are conventionally derived rigorously in elementary calculus classes, and no way of computing the area of either (without knowing the area of at least one of them) avoids the use of calculus of some sort.(By the way, "calculus" is not a proper noun, don't treat it as such)--Jasper Deng (talk) 20:17, 19 October 2016 (UTC)[reply]

Calculus exists, no one argues about that. But the area formula of the circle was derived thousands of years ago, even if our 350 (or so) year-old calculus had "confirmed" it. Rigorous or not, People used it ever since.

But I realized we ran of topic ― my main question is why isn't this non-rigorous method been taught in schools and other places as well as the circle area non-rigorous method? Until I found that website page all I could find were rigorous calculus proofs. יהודה שמחה ולדמן (talk) 20:57, 19 October 2016 (UTC)[reply]

I feel like there's really not much difference between stretching and calculus here. I mean, to "stretch" a circle on one dimension means that for any given rectangular element with a large dimension y and a small dimension delta-x, the length of delta-x is altered accordingly. In the limit as delta-x approaches zero, the rectangular elements cover the entire ellipse, while retaining the same degree of stretching; hence, the area of the ellipse is altered accordingly. I feel like there must be some general theorem to this effect, regarding any shape, which any real mathematician could name. Wnt (talk) 21:21, 19 October 2016 (UTC)[reply]

(edit conflict) It is taught here in the USA. It's just that the ellipse itself has much less pedagogical value than the circle (yes it's a conic section, but the circle is best for teaching trigonometric functions). And at least here, non-rigorous methods are not actually mentioned that much in general.

Strictly speaking, the formula wasn't derived until it was first rigorously proven. The method of circum- and in-scribing polygons implicitly assumes that the circumference of the circle lies between those of the polygons, and the least upper bound property of the reals - assumptions that the Greeks did not have the tools to address rigorously.--Jasper Deng (talk) 21:22, 19 October 2016 (UTC)[reply]

I thought Eudoxus of Cnidus had proved the circle area formula by contradiction, no? יהודה שמחה ולדמן (talk) 22:20, 19 October 2016 (UTC)[reply]

As far as I know, he only established that the area is proportional to r² (which can now be easily proven using Fubini's theorem), and did not establish the constant of proportionality. Given that pi most certainly is an "incommensurable" quantity, and that he didn't like dealing with those, I doubt he would have proven the whole formula. Even if he did, the assumptions I mentioned above are still implicit in his method.--Jasper Deng (talk) 23:06, 19 October 2016 (UTC)[reply]