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The square's sides will be the sqrt 2 = about 1.414 inches.
(NOTE that this only happens when the circle is 2 inches in diameter. For a circle 3 inches in diameter, the diagonal measurement 3 = (sq rt 2) s and s = about 2.122 inches, not the square root of 3)
The largest square will have a DIAGONAL the same length as the DIAMETER of the circle. This is clear when you imagine that it is the largest dimension of any square. With the diagonal length being the hypotenuse of the two right triangles that form the square, the diagonal length equals the square root of the sum of the squares (of the two sides of the square)
diagonal = (sq rt 2) x side
diameter = sq rt (s^2 + s^2).
2 inches = sq rt (2 s^2)
2 inches - (sq rt 2) x s
2 inches = 1.414 s
and s = 1.414
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