What is the basic idea behind calculation of area?

Question

The system of calculating area in terms of square units is pretty philosophical and not very intuitive. It must have taken a great amount of time for humanity to arrive at such a convention and to spread it across different societies. My question is about finding basics of such a convention and should the person who first thought of calculating areas in square units be regarded a great philosopher equivalent to Newton and Einstein? Moreover, could we have evolved a different method of calculating areas?

Answer 1

Area is a property of a planar region. The formulation in terms of "unit area" comes from the fact that many of the properties of planar regions that human beings care about are translation-invariant. For example, how much grain or how many sheep a region of land can support, or how long it will take one person to till it.

All of these quantities have two important properties:

1. Translation invariance. If $A$ and $B$ are the same shape and size but in different places, then $f(A)=f(B)$.
2. Additivity. If $A$ and $B$ are disjoint regions, then $f(A\cup B)=f(A)+f(B)$.

These two properties were presumably intuitively obvious to ancient humans, without them having to formulate them precisely or even consciously. From these properties it follows that if you take any given shape $U$, then for any region $A$, $f(A)$ is determined by $f(U)$ and by how many disjoint copies of $U$ it takes to tile $A$. Therefore, how many copies of $U$ it takes to tile a region is a good measure of the "size" of land. It's surely visually very obvious that any region can be reasonably tiled with small squares.

Of course, all this is just a very complicated, modern way of saying something very simple: since any given 1x1 meter square can contain the same amount of wheat, well, obviously if I know how many square meters cover a given region of land, I know how much wheat I can put on it.