Area of Square

Theorem
A square has an area of $$L^2$$ where $$L$$ is the length of a side of the square.

Integer Side Length
In the case where $$L=1$$, the statement follows from the definition of area.

If $$L \in \mathbb{N}, L > 1$$, then we can divide the square into smaller squares, each of side length one.

Since there will be $$L$$ squares of side length one on each side, it follows that there will be $$L\cdot L = L^2$$ squares of side length one.

Thus, the area of the square of side length $$L$$ is $$L^2 \cdot 1 = L^2$$.

Rational Side Length
If $$L$$ is a rational number, then $$\exists p,q \in \mathbb{N}: L=\frac{p}{q}$$. Call the area of this square $$S$$.

We can create a square of side length $$c = L \cdot q$$, and we call the area of this square $$S'$$. We then divide this square into smaller squares of side length $$L$$.

Since there will be $$q$$ squares of side length $$L$$ on each side of the larger square, it follows that there will be $$q^2$$ squares of side length $$L$$.

Thus, $$S' = q^2 \cdot S$$.

From the integer side length case, $$S' = c^2$$.

So $$L^2 \cdot q^2 = (L\cdot q)^2 = c^2 = S' = q^2\cdot S$$.

Finally, $$L^2 = S$$.