So, today I learned that the area of a circle is $\pi r^2$. So, I thought that since $r$ is $1$ dimensional, $r^2$ will be $2$ dimensional. In this case, a square, as you only multiply $2$ dimensions (without additional manipulation to change the shape).

But then, what does $\pi$ do to the square? How can a square become a circle with $\pi$?

Possible answers that I thought are that the area of the circle is equal to $3.1415\ldots$ squares (with $r$ side length). And that the formula $\pi r^2$ is derived from a long formula(I would like to know the long formula if this is true, because how do mathematicians get the area of the circle before comparing in the first place?)

I asked my teacher about this but he can't really understand me. So, I hope experts at StackExchange understand my problem.

[A picture I drew, showing that $r^2$ is a square](https://i.imgur.com/WWpAZOF.jpg)