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Square in the Circle

There is a circle from within which a square is cut, the remaining portion having an area of 47.6255 square units.   The diameter of the circle is 7 more units than the square root of the side of the square.  Find the diameter of the circle and the side of the square. [Note: this problem uses the approximation \(\pi \approx 3.142\).]

Kongenki, Sato Seiko, 1666

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"Square in the Circle," Convergence (March 2007)