$8, \sqrt{1+\frac{\sqrt{2}}{2}}$

Explanation: Extend the four sides of the square $L I J K$ to intersect the circle at points $A B C D E F G H$, as shown in the figure. To be able to see two complete sides of the square $L I J K$, the point must lie on one of the four minor arcs: arc $A B$, arc $C D$, arc $E F$, or arc $G H$. Since the probability is 0.5, according to the geometric probability model, arc $A B$ should be $45^{\circ}$. Because $\angle A I B=90^{\circ}$, the circle with center $I$ and radius $A I$ passes through the center $O$ of the square $L I J K$. Also, $\angle A I O=135^{\circ}$, and $I O=\frac{\sqrt{2}}{2}$, hence
$$
A O=\sqrt{1+\frac{\sqrt{2}}{2}} .
$$
8、A unit square has its center coinciding with the center of a circle and is located inside the circle. If a point is randomly selected on the circle, and the probability that from this point one can see two complete sides of the square is $\mathbf{1 / 2}$, then the radius of the circle is $\qquad$