Where did we get $\pi r^2$ from? [closed]

Question

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 $\pi$.

Answer 1

$\pi$ 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:

Thanks to Eloquent Math for the picture.

Answer 2

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, $$\left(\frac{1}{2}br\right)N = \left(\frac{1}{2}r\right)bN \to \left(\frac{1}{2}r\right)2{\pi}r = {\pi}r^2.$$

Answer 3

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.

Answer 4

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 $\pi$.

Answer 5

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.