why do we measure area in square units

I was recently asked "Why is the area of a circle irrational? ", to which I replied that it was not necessarily irrationalБthere 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? If you want to measure area, you need to use a flat shape.

Think about what measuring means: comparing this thing we are interested in to a similar thing that we've agreed to use as a unit. To measure length, you need to compare it to some other length, to see whether they mark off the same amount of distance. To measure volume, you compare it to some other volume, to see whether they hold the same amount of air/water/ether/whatever. To measure area, you compare your shape to some other flat shape to see whether they cover the same amount of flat space. Now, what will be a convenient unit for counting off the amount of flat space something covers? Our unit should be: symmetrical, so it doesn't matter which way we lay it down space-filling, so we don't miss some part of the area we want to cover easily imagined or constructed, so we don't have to put too much effort into making a measurement The first condition limits us to circles or.

The second condition eliminates the circle and limits us to polygons that can. Of these, the square is by far the easiest to work with. These considerations do not depend on the system of measurement you are using (inches, cm, or cubits), but only on the nature of flat shapes. I don't know of any civilization that developed any other way of measuring flat space, and I doubt there ever was one. Even triangles, which seem like they should be a simpler shape than squares (having only three sides instead of four) are much more difficult to use as a measuring system, because it is harder to create fractions of triangles, and you know that measurements do not always come out in nice, even numbers.

Edited to add: I've read somewhere that Descartes did not originally insist that the axes of his coordinate geometry system be perpendicular. We are used to working with perpendicular axes, but everything would work just as well with a slant-wise system. (Try it and see! ) In that case, I suppose we could measure area in rhombuses. I think it was the ease of working with squares that pushed history into using the perpendicular coordinate system we have today.