Why is the area of the circle $πr^2$? [duplicate]

I searched many times about the cause of the circle area formula but I did not know anything so ...

Why is the area of the circle $\pi r^2$?

Thanks for all here.

How Archimedes viewed it:

$%emptyuselesstextemptyuselesstextemptyuselesstext$

As the width of the slices approaches $0$, the object on the right-hand side approaches a rectangle of width $2\pi r /2 = \pi r$ and height $r$, hence area $\pi r^2$.

The area of a circle with radius $r$ is just $r^2$ times the area of the unit circle, by homothety. So the area of the circle is the square of the radius times a universal constant, given by: $$\begin{eqnarray*}2\int_{-1}^{1}\sqrt{1-x^2}\,dx &=& 4\int_{0}^{1}\sqrt{1-x^2}\,dx = 2\int_{0}^{1}x^{-1/2}(1-x)^{1/2}\,dx\\ &=& 2\frac{\Gamma(1/2)\Gamma(3/2)}{\Gamma(2)} = \color{red}{\Gamma(1/2)^2}.\end{eqnarray*}$$