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This is basic question. Since childhood I am mugging the mathematical formulae areas of square, rectangle and circle etc.

Now,it is possible for me to understand formula of area of square I.e. square of length, area of rectangle as product of length and breadth.

But I still don't understand formula for area of circle I.e. $A=\pi r^2$

I want to know that

  1. How do mathematicians invented the "$\pi$" term?

  2. How do they introduced "$\pi$" term in the case of the circle? (circumference and area of the circle)

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    $\begingroup$ If you are only interested in the constant $\pi$ you can get to it like this. Imagine you have a circle and you are able to measure its circumference "c". Then, you can also find out what its diameter "d" is. When you divide circumference by diameter, that is "c/d" you get a number. When you try this on various circles, and if you measure "c" and "d" more and more precisely, you will see that you get some constant, that is called $\pi$. To be mathematically correct/formal, you also need to prove a bit ;-) $\endgroup$ – Jan Jun 11 '15 at 11:27
  • $\begingroup$ @Jan actually it's not %pi, it's $\pi$, rendered as $\pi$ ;) $\endgroup$ – Ruslan Jun 11 '15 at 11:28
  • $\begingroup$ Sorry, I have been working in Scilab and there we have %pi :-) $\endgroup$ – Jan Jun 11 '15 at 11:29
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The ancient Greeks (in particular, Eudoxus) proved that circles have areas that scale with the square of the radius, i.e. if the circles have radii $r_1$ and $r_2$, the ratio of the areas is $$ \frac{A_1}{A_2} = \frac{r_1^2}{r_2^2}. $$ So if I have a circle of radius $1$, it has area equal to some number, say $a$. Then I can find the area $A$ of a circle of any radius $r$ by using the formula above as $$ \frac{A}{a} = \frac{r^2}{1^2} \\ A = ar^2. $$ In particular, we now refer to the constant $a$ by the Greek letter $\pi$.

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  • $\begingroup$ How did they prove that $\frac{A_1}{A_2}=\frac{r_1^2}{r_2^2}$? $\endgroup$ – maxuel Jun 11 '15 at 11:32
  • $\begingroup$ See references on the wikipedia article : en.wikipedia.org/wiki/Area_of_a_disk $\endgroup$ – Nihl Jun 11 '15 at 11:40
  • $\begingroup$ By contradiction. This page seems to have the technical details. It is suspected that Euclid's version is neater than Eudoxus's but we don't know since this is the earliest we have (unless Archimedes is earlier than Euclid). Essentially you take an inscribed polygon, double the number of sides, and show that more than half the area between the polygon and the circle is removed. You do the same thing on the outside (Euclid doesn't, because it would be more complicated to formulate formally), and hence find that they cannot be... $\endgroup$ – Chappers Jun 11 '15 at 11:42
  • $\begingroup$ ... not as the ratio of the squares on their diameters. @maxuel $\endgroup$ – Chappers Jun 11 '15 at 11:42
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Given: The circumference of a circle is $2\pi r$

To prove: The area is $\pi r^2$

Circle area proof

Cut the circle into lots of sectors ("pizza slices"), and rearrange into an almost-rectangle. The more slices you use, the closer you get to an actual rectangle. This rectangle has width $\dfrac C2=\pi r$ and height $r$ (why?). So the area of the circle is $\pi r\times r=\pi r^2$.

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  • $\begingroup$ Search "area circle proof" on Google images for more pictures like this. $\endgroup$ – Akiva Weinberger Jun 11 '15 at 13:37
  • $\begingroup$ @c thanks for the explanation but how you measure circumference of circle? $\endgroup$ – sagar Jun 11 '15 at 13:45
  • $\begingroup$ Does this GIF help you? $\endgroup$ – Akiva Weinberger Jun 11 '15 at 14:10
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Imagine you have a circle of diameter $1$ on the floor and you mark the point where the circle and floor make contact. Roll the circle until the point of contact touches the floor again. If you measure the distance traveled the distance will be $\pi$. This is true for any radius/diameter in proportion.

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