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For any given perimeter, the circle contains the largest area. In 3-d, a sphere will contain the largest volume. Back tro 2-d, the increase in area you get from straightening out a circle is less than the amount of are that you lose.

In terms of formulae:

circumference = 16 = 2*pi*r

so r = 8/pi

Then area = pi*r2 = pi*64/pi2 = 64/pi = 20.372 sq units

Area of a square with sides 4 = 4*4 = 16 sq units

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Q: Why is the area of a circle with a circumference of 16 greater than the area of a square with sides of 4?
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