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.

The simplest explanation is that the area of any shape has to be in units of area, that is in units of length squared. In a circle, the only "number" describing it the the radius $r$ (with units of length), so that the area must be proportional to $r^2$. So for some constant $b$,
$$A=b r^2$$
Now, to find the constant $b$, I think the easiest way is to look at this Wikipedia diagram:

This shows how when you subdivide the circle into many equal small triangles, the area becomes a rectangle with height $r$ and length equal to half the circumference of the circle, which is $\pi r$, by the definition of $\pi$.
