I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula of parallelogram, then triangle and so on. That brings us to the question how to prove that the area of the square with side $a$ is $a^2$.
I just came upon ProofWiki article (http://www.proofwiki.org/wiki/Area_of_Parallelogram/Rectangle), which showed me to get the area of rectangle from that of the square, which completes the rest of the link. But the square?
I see that for such a basic figure we need some axioms to get our foot down. One that I think is absolutely essential that that of a square of $1$ units has an area of $1$ unit square, from which we can generalize to bigger squares. Another I think which we used in the rectangle proof that, if a figure $A$ is divided into different pieces then the area of $A$ is the sum of the area of the different pieces.
So my first question: what set of axioms are needed that best and unambiguously describe the area of figures?
Anyway, the one informal proof we are show in primary schools, that divided for example a $5 \cdot5 $ square into $25$ pieces of $1$ cm$^2$ (intuitively very plausible) fails to convince me. Somehow, we may generalize it to rational numbers, but what if the side is $\pi$? We cannot keep dividing it infinitely many times.
So the second question: How do we using the axioms find the area of the square? And is there any gap, when we move forward, for example with the rectangle? Thanks.