What does "area" really mean? My professor had an interesting statement at the beginning of first year integral calculus.

What does area really mean? 

How do we know that the area of a circle is $\pi r^2$? Archimedes used the method of exhaustion.
Is there a good generalisation for the meaning of area with respect to how it concerns integral calculus? I apologise if the question is either too general or just stupid. But it's interesting, and I would appreciate more interesting examples on the nature of area in general. 

A naive approach: Pick a plane region and fill it with as many circles of radius $1$ as you can (without overlapping), and then count them all. Let the result be $a_0$. After that, pick circles of radius $r/2$, count them all and divide the result by 2. Let this result be $a_1$. Iterate the process, so every time you fill with circles of radius $2^{-n}$, count them all and divide the result by $2^n$, letting this value be $a_n$. Then, the area of the region can be thought as the limit when $n \xrightarrow[]{} \infty$ of the sequence $\left\{ a_n \right\}$. 
