jmc

The diagonal of a square is the hypotenuse of a 45-45-90 right triangle whose legs are the sides of the square. Therefore, the diagonal of the square measures $12\sqrt{2}$ units. The diagonal of the square is a diameter of the circle, so the radius of the circle is $6\sqrt{2}$ units. The area of the circle is $\pi (6\sqrt{2}{)}^2=72\pi$, so $K=\boxed{72}$.