There are many different ways to cut and rearrange the pieces of a rectangle into a square. The least amount of pieces that you would need to cut your rectangle into would be 2.
If you start with a rectangle with dimensions of the ratio 4:1, you can cut it in half vertically and put one half on top of the other to make a square. This is the only way that you can make a square out of a rectangle by cutting it into only 2 pieces.
There are also a lot of ways to rearrange the pieces of a rectangle into a square by cutting it into 4 pieces. One way that you can do this is by cutting a rectangle with the ratio 2:1 into half vertically. You can then cut each of these squares into half diagonally. Next, rearrange them to make a square.
From this experiment I have learnt that the area of a rectangle is equal to the area of the square that the pieces have been rearranged into. I have also concluded that you can cut up and reassemble the pieces of a rectangle into a square by using finitely many pieces.