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Start with a square. You already get this.
How much floor does a shape cover? For a square, the answer is the square itself.
Count one side. Say it is 3. Then you get 3 rows of 3 tiles. That is 9. Two of the same length, multiplied. People write it r × r, or r² — not as decoration. It is literally the square. Every other area trick in this story is trying to get back to something this obvious.
Drag the corner. When the side is a whole number, count the tiles.
In your life
A chessboard, a window, a slice of pizza box. Any time you ask “how much floor,” you are doing this. The rest of math is trying to get back to a square you can count.