... but I can't figure out why???
Area to perimeter ratio.
Circle area = pi r^2 perimeter = 2 pi r ratio = r/2
Square area = (2r)^2 = 4 r^2 perimeter = 4 x 2r = 8 r ratio = r/2
But the circle has the highest ratio of area to perimeter! That's a well known fact.
Another proof shows with equal perimeters the areas of a circle and a square are in the ratio of 4/pi.
One of these must be wrong. I'm sure it is the calculations above, but I'm not seeing it. I have to be messing up an assumption.