Asquare is inscribed in a circle with an area of 10 pi square inches. what is the area of the square

We need to determine the radius and diameter of the circle.  If the area of the circle is 10 pi in^2, then, according to the formula for the area of a circle,

A = 10 pi in^2 = pi*r^2.  Thus, 10 in^2 = r^2, and r = radius of circle = sqrt(10) in.

Thus, the diam. of the circle is 2sqrt(10) in.  This diam. has the same length as does the hypotenuse of one of the triangles making up the square.

Thus, [ 2*sqrt(10) ]^2 = x^2 + x^2, where x represents the length of one side of the square.  So, 4(10) in^2 = 2x^2.  Then:
                                 40 in^2  = 2x^2, or 20 in^2 = x^2, and so the length x of one side of the square is sqrt(20).  The area of the square is the square of this result:

Area of the square = x^2 = [ sqrt(20) ]^2 = 20 in^2 (answer).  Compare that to the 10 pi sq in area of the circle (31.42 in^2).