TITLE: Why is the area of the circle $πr^2$?
QUESTION [16 upvotes]: 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.

REPLY [2 votes]: For another integration method, let $C$ be the interior of the circle $x^2+y^2=r^2$ with radius $r$. Then the area, with the aid of polar coordinates, is given by 
\begin{align*}
\iint_C dA &= \int_0^{2\pi}\int_0^r\rho\,\mathrm{d}\rho\,\mathrm{d}\theta\\
&=\frac{1}{2}r^2\int_0^{2\pi}\,\mathrm{d}\theta\\
&=\pi r^2. 
\end{align*}