11. Given that the radius of $\odot O$ is $1$, and $P$ is a point on the circumference. As shown in Figure 3, a square with side length $1$ (indicated by the solid line, with vertex $A$ coinciding with $P$) rolls along the circumference in a clockwise direction. After several rolls, point $A$ returns to point $P$ for the first time. The length of the path that point $A$ has traveled is $\qquad$
11. $\frac{2+\sqrt{2}}{2} \pi$.

Point $A$ travels a path consisting of nine minor arcs, each with a central angle of $\frac{\pi}{6}$. Six of these arcs are from circles with a radius of 1, and three arcs are from circles with a radius of $\sqrt{2}$. Therefore, the length of the path traveled by point $A$ is
$$
\frac{\pi}{6}(6 \times 1 + 3 \times \sqrt{2}) = \frac{2+\sqrt{2}}{2} \pi .
$$