Long ago, folks didn't know about pi, and they didn't know how to find
the area or circumference of a circle. So they came up with methods that
they thought approximated those things.
Take a circle. Circumscribe a square around the circle and inscribe a
square in the circle. Orient both the squares the same way. Now construct a
square exactly in between the two squares.
It was believed that this "middle" square had the same area as the
circle, and that its perimeter was equal to the circumference of the
circle. Is this true? Be sure to explain your answer thoroughly, and
mention anything interesting that you notice along the way.