How do mathematicians invented and introduced $\pi$ term in the case of circle?

Question

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)

Answer 1

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 $\pi$.

Answer 2

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.

Answer 3

Given: The circumference of a circle is $2\pi r$

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

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 $\pi r\times r=\pi r^2$.