Why is the area of a square $A = s^2$?($s$ is the side length)

Question

Can some provide a proof for the formula $A = s^2$ for the area of a square?

Answer 1

$$A = \int\limits_{x=0}^s \int\limits_{y=0}^s 1\ dx\ dy = s^2$$

Answer 2

This depends a lot on your assumptions about area, and there are probably a handful of very different approaches.

Here is one: suppose we want to define "area" for at least some subsets of the plane $\mathbb R^2$, like the ones we would call "squares". There are some properties we might find intuitive, like:

1. The area of a set (when it exists) is always a nonnegative real number or $\infty$. (Maybe we want to say things like "the area of the plane is $\infty$.")
2. The area of the empty set is $0$.
3. The complement of something with area has area.
4. The intersection of two things with area has area.
5. The area of (your favorite version of) the unit square is $1$. (If we don't pin down the area of anything with nonzero area in particular, then we could accidentally define something like "twice area" by mistake.)
6. If you translate/shift a set around in the plane, its area shouldn't change.
7. If you break a set $S$ into (countably-many) disjoint subsets $S_1,S_2,\ldots$, each of which has an area (possibly $0$ or $\infty$), then the area of $S$ should be the sum of all of the areas. We use sometimes ideas like that when working with integrals or thinking about complicated areas like that of a circle.
8. For convenience, any subset of a set with area $0$ should definitely have an area; and by the above property, that area is forced to be $0$. (Intuitively, one of those subsets should have area $0$ since it's just a piece of something with area $0$.)

With fixed assumptions about set theory, the above assumptions pin down the notion of area in the plane uniquely. There are no more choices to be made. The name for this concept in arbitrary numbers of dimensions (length in the real line, area in the plane, volume in 3D, etc.) is the Lebesgue measure, and it's usually first studied in depth at or near the graduate/post-grad level.

From here, proving that the area of any square is what it should be can be done in a few ways, but you'll need to use infinite sums/limits somewhere if you want more than something like "half-open rational-side-length squares have the area they should if they have area at all" (taking assumption 5 to mean "$[0,1)\times[0,1)$ has area $1$").