Why is $\pi r^2$ the surface of a circle Why is $\pi r^2$ the surface of a circle? 
I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why this formula works. 
So I want to understand why it works and not just how.
Please don't use complicated symbols.

One way to prove it is the following : Embed a regular polygon (ie. all sides are equal, all angles are equal) in a circle (ie. its vertices are on the circle. Divide this polygon into $n$ triangles by drawing radii out to the vertices. Calculate the area of this triangle and multiply by $n$, and you will get the area of the polygon.
Now, if you can assume the following limit :
$$
\lim_{x\to 0} \frac{\sin(x)}{x} = 1
$$
Then when you take a limit as $n$ goes to infinity, you will end up with $\pi r^2$.
So all that remains is proving the above formula :)
