The local area

Everyone knows that if you have a square with sides each of 2cm, the area is 4cm², or is it?

Everyone knows that if you have a shape, its area does not change. if you make a shape out of shapes, then the bigger shapes area does not change either... no matter how you re-arrange the bits...

Justifying this another way... If you have a large area, and cut it into smaller parts, the area you get when you put it