I was messing around with geometry a bit today and came across a problem that I do not understand.

I created a circle with a radius of 2. The circumference of such a circle is 12.56, and the area is 12.56

Now...I tried to 'fit a square' into the circle, but not in a typical way. I decided to bend the square to fit perfectly into the circle with no space left over...effectively creating a 4 'sided' circle. Now...each 'side' of the circle/square is 3.14, or pi, since 12.56/4 = pi. Now...if you try to find the area of this circle-square using the formula for the area of a square, you get 3.14^2 = 9.86

This value is different from the 12.56 of the traditional area of the circle formula. The problem I have with this is that no matter how you re-arrange the sides of a square, if the perimeter is the same, the area will be the same. So...why is it that when you re-arrange how the perimeter is 'counted' from a circle to a square, the area changes?