For a square inscribed in a circle with radius r, the diagonal of the square is equal to the diameter of the circle.

Let the side of the square be s, its diagonal by the Pythagorean Theorem is given by sqrt(s^2 + s^2) = sqrt (2*s^2) = s*sqrt 2.

The diameter of the circle is 2r.

Equating the diameter and the diagonal we get:

2r = s*sqrt 2

=> s = 2r/ sqrt 2

=> s = r*sqrt 2

The area of a square with side equal to s is s^2. The area of the square here is s^2 = (r*sqrt 2)^2 = 2*r^2

**The expression for the area of the square in terms of r is 2*r^2**.