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The length of the side of the square is a.

$r^2=2\cdot (\frac{a}{2})^2\\ 1=2\cdot \frac{a^2}{4}\\ a=\sqrt{2}\\ A=a^2$

$A=2$

The area of the largest square

that can be inscribed in a circle with radius 1 is 2 square units.

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