None unless you draw some inside.
^ Terrible answer:
There can be many different numbers of squares inside a circle. As the size of the squares goes to zero, the number of squares goes to infinity.
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A cardioid is the line a point on the edge of a circle traces when the circle is rolled around a circle of equal radius. A nephroid is the line a point on the edge of a circle traces when the circle is rolled around a circle of twice the radius.
US Army Pin or Patch The Discharge Pin was a diamond shaped pin with an eagle inside a circle, just as you described. The patch was yellow embroidered on tan or brown. The pin was an all Brass pin. This was commonly referred to as the "ruptured duck" pin because the eagle in the insignia looks more like a duck. These pins or patches were awarded to soldiers who were discharged and it allowed them to wear the pin on their uniform and continue to wear their uniform until they were able to get home. This allowed them some relief from certain duty and orders. Link (scoll down to see Discharge Pin):http://custermen.com/Reference/Uniforms/Badges.htm#Medals&Insignia
to construct (using a compass and straight-edge) a square with the same area as a given circle using only a finite number of steps. "Squaring the circle" was an ancient problem that has been proved impossible to do.
You have to determine the value of pi squared.