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I'm wondering if someone can explain how we got the formula for the area of a circle to be $\pi r^2$, and perhaps even more precisely, how we came up with $\pi$.

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closed as not a real question by Pedro Tamaroff, Jennifer Dylan, draks ..., rschwieb, Ayman Hourieh Oct 19 '12 at 14:00


This question is very vague. What does " how we came up with $π$." mean? "how we got the formula for the area of a circle to be $π$" has also very different possible answers. Could you be more specific? – Pedro Tamaroff Oct 19 '12 at 2:19
There are two questions; How did someone derive the formula for the area of a disk, and where did he get pi from? – Imray Oct 19 '12 at 2:29
Is the question how we found out that there exists a constant $c$ such that $A=cr^2$? Or how we found out that this $c$ equals the quotient of circumference and diameter? Or how we found out that it equals half the absolute value of the period of all nonzero functions that are their own derivative? Or just why we use the letter $\pi$ instead of $c$? – Hagen von Eitzen Oct 19 '12 at 14:13

5 Answers 5

up vote 5 down vote accepted

$\pi$ was found by early mathematicians by finding the ratio of circumference to diameter. Check out Wikipedia for more detail.

As for the formula $a=\pi*r^2$, check out this picture: enter image description here

Thanks to Eloquent Math for the picture.

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shell integration

Wikipedia actually does a fantastic job of explaining, with several proofs. For example, the animation above demonstrating area via shell integration is from the page linked.

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Wiki does not prove that the process is area preserving. – zyx Oct 19 '12 at 2:41
@zyx I doubt many people regard the animation as a rigorous proof. It's more of a heuristic aid in my opinion. – EuYu Oct 19 '12 at 2:52
Of course, but it would help if the animation had different colors for points that start along some particular radii, to show that the layers do not stretch and slide relative to each other. – zyx Oct 19 '12 at 2:57
@zyx The beauty of Wikipedia is that you could make that change. I encourage you to do so. – Quinn Culver Oct 19 '12 at 3:02
@Quinn, do you know how the animations are done? I don't, but would be happy to learn. – zyx Oct 19 '12 at 3:17

For question " from where did we get the Pi", following is the copy paste from the book "Pi: a biography of the world's most mysterious number" by Alfred S. Posamentier, Ingmar Lehmann:

Perhaps in the early days it was important to measure how far a wheel would travel in one revolution. This might have been done by rolling the wheel on the ground and marking off the distance it rolled in exactly one revolution (without slippage, of course) or with some- thing resembling a string placed along it. The diameter, a much easier dimension to measure, since it merely required placing a straight stick or rule alongside it and marking off its length, was probably also noted. We can assume that these two measurements were compared for various circular objects. This was likely the beginning of the establishment of comparison between the two measurements that seem related to each other.

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EDIT: (The following speculative paragraph, especially in light of @MJD's comment, might very well be wrong.) Probably, ancient people had measured the circumference and diameter of circles they'd drawn (perhaps with compass and ruler) and noticed that the ratio $c/d$ was about 3.14 every time. This might have been initially puzzling to our ancestors, but to us in modern times, it's very clear why.

Even if the previous paragraph is incorrect, we can still understand why $\pi$ is a natural thing to consider. The theory of similarity in geometry tells the tale. Intuitively two shapes are similar if they have the same shape, but one is bigger than the other. Another way of saying it is that one of them is 'scaled down' to get the other. (So if someone shows you a scale drawing of something, the drawing and the original something are similar.)

For example, draw a triangle with side lengths $3$, $4$, and $5$, and then another with side lenghts $9$, $12$, and $15$. You'll notice that they have the same shape but the latter is bigger. It is always true (for similar figures) that the ratio of two parts (of sides) of one is the same as the corresponding two parts in the other. For example, $3/4 = 9/12$ and even $3/(4+5) = 9/(12+15)$.

Of course, all circles have the same shape and hence all circles are similar. Because they're similar, the circumference-to-diameter ratio $c/d$ is constant from circle to circle. Someone decided to call that constant value $\pi$.

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Ancient peoples could neither measure with enough precision nor draw circles with enough accuracy to determine the value of π to one part in 300 as you suggest. Early determinations of π to that degree of precision were made by theoretical calculations, not by direct measurement. – MJD Oct 19 '12 at 4:19
@MJD Interesting. Do you have a reference? Notice that the "one part in 300" might not be crucial to my hypothesis though, since even if they noticed that it was about 3 each time, they still might've guessed that it was constant. – Quinn Culver Oct 19 '12 at 13:56
My complaint isn't that they didn't notice it was constant; I think the approximation $\pi\approx 3$ has been known since prehistoric times. My objection is only to the "3.14" part of the claim. Wikipedia's article on Approximations of π has a history of approximations to π. In particular, the Babylonians were using the value 25/8 = 3.125 about 3800 years ago, which they would not have done had they been able to straightforwardly measure that it was closer to 22/7. – MJD Oct 19 '12 at 16:31

Place isosceles triangles, of equal area, inside a circle. Then the circle's area can be approximated by the area of all the triangles. If $r$ is the radius, $b$ is an arbitrary base of each triangle, and $N$ is the number of triangles, then the area of all the triangles is given by $$\left(\frac{1}{2}br\right)N.$$ However, observe that as we add more triangles, and make each triangle smaller, $bN$ approaches the circumference of our circle, that is, $$\left(\frac{1}{2}br\right)N = \left(\frac{1}{2}r\right)bN \to \left(\frac{1}{2}r\right)2{\pi}r = {\pi}r^2.$$

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