I was recently asked "Why is the area of a circle irrational?", to which I replied that it was not necessarily irrational—there are of course certain values for $r$ that would make $\pi r^2$ rational. She proceeded to clarify, "But the area of a square of side length $1$ is rational, yet the area of a circle of radius $1$ isn't. What's so special about the square?"
My answer to this was of course "We measure area in square units, hence the area of a unit square is one square unit." Perfectly contented, I headed home. On the way back, however, it dawned on me that this was unsatisfactory. Why must we measure area in square units? Area is one of those quantities that one could scale by any constant, such as $1 \over \pi$, and have almost every property preserved. Is there a fundamental reason why we don't say the area of a unit square is $1 \over \pi$ circular units?
This further confused me when I noticed that the phrase $x^2$ being spoken as "x squared" is a consequence of using the unit square, not a justification for it. If the Ancient Greeks used circular units, we would no doubt pronounce $x^2$ as "x circled"...
I looked for a justification for using the unit square as the basis unit for area, and of course the obvious one is calculus. The Fundamental Theorem of Calculus provides an easy definition of area, thanks to integrals. The area of a unit square is simply:
And since the integral is the antiderivative, it is convenient to say the area of a unit square is $1$.
But I am still not sure this is unsatisfactory, as the intuitive connection between integrals and area relies on the concept that the area of a rectangle is $lw$. Had this been off by a factor of $1 \over \pi$, the connection between antiderivatives and areas would certainly be more complicated... but many mathematical formulae have $\pi$ or $1 \over \pi$ in them, and it would be a stretch to proclaim them inelegant.
In the end, is there something fundamental about the unit square? Why not a unit triangle or unit circle? Or is it so merely because the Ancient Greeks did it that way?