jmc

The diagonal of a square inscribed in a circle is a diameter of the circle, so the length of the diagonal of the square is 2 units. Recall that the area of a square with diagonal $d$ is $d^2/2$. The area of a square whose diagonal measures 2 units is $2^2/2=\boxed{2}$ square units.

Note: To derive the area formula $d^2/2$, divide the square into two 45-45-90 right triangles. The side length of the square is $d/\sqrt{2}$ and its area $(d/\sqrt{2}{)}^2=d^2/2$.