Because the radius of the circle is 10 inches, that means that half of the diagonal of the square is 10 inches. (we can tell this because the square is inscribed in the circle, and so the diagonal of the square is the diameter of the circle). This makes the whole diagonal 20 inches, and each side of the square being 20/sqrt(2) or 10sqrt(2) inches. (We know this due to 45-45-90 triangles, which state that if a 45-45-90 triangle or a right isosceles triangle has a leg a, then the hypotenuse has to be a*sqrt(2)). (10sqrt(2))^2 gives us the area of the square, or 200 in^2.
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