Why square units?

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:

$$\int_0^1 dx$$

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?

Filling areas with circles is a futile endeavour, since there's always a free space left over in between neighbouring discs. Furthermore, there's a direct connection between the geometric shapes called rectangles, and multiplication, inasmuch as the latter's commutative property becomes self-evident when one pictures a $m\times n$ grid, which obviously holds as many elements, then rotates it at a $90^\circ$ angle, thus transforming it into an $n\times m$ grid, which obviously holds just as many elements as before being rotated, since it is still the same grid. So there are both practical as well as theoretical advantages to using rectangular shapes.

If you want to know the area of something in terms of how many copies of some standard shape cover that area, the standard shape needs to be something that tiles the plane in a tessellation. Otherwise -- for example, if you use a unit circle or a unit regular pentagon -- when you try to cover a given area with a shape that cannot tile the plane, for most areas you will inevitably have gaps between the unit shapes or places where the unit shapes overlap or both.

The only regular polygons that tile the plane are the regular hexagon, the square, and the equilateral triangle.

(I suppose you could specify that areas are covered by close-packed circles, but each of those circles corresponds to a regular hexagon, which together completely cover the same area without overlaps or gaps in a hexagonal tiling, so you might as well use a hexagon as your unit of area).

Some people measure the area of geometric shapes in units of "unit tetras" or "unit triangles" instead of the classic "unit cubes" or "unit squares".

Likewise, if you want to know the volume of something in terms of how many copies of some standard shape fill that volume, the standard shape needs to be a space-filling polyhedra that fills space to form a honeycomb.

(I suppose you could specify that volumes are filled by close-packed spheres, but each of those spheres corresponds to a rhombic dodecahedra, which together completely cover the same area without overlaps or gaps in a rhombic dodecahedral honeycomb, so you might as well use a rhombic dodecahedron as your unit of area, like honeybees do).

The only Platonic solid that is space-filling by itself is the cube. However, Buckminster Fuller shows that sometimes it is convenient to use the regular tetrahedron with edge length 1 as a unit of volume. The tetrahedral-octahedral honeycomb fill space, as in a octet truss, composed of regular octahedrons of edge length 1 (which has a volume of exactly four unit tetras) and unit tetras.

Geometry comes from geo-metry, earth measuring. The original motivation behind geometry was to define "this is my land, that is yours, here is the border, and mine is bigger". For this to work, two requirements must be met:

• Two adjecent fields must be connected without "loss", like the star-shaped space between circles.
• A border must be easy to define, like a straight line between two landmarks.