6. As shown in Figure 1, the side lengths of the equilateral triangle and the square are both $a$. In the plane of the figure, $\triangle P A D$ is rotated counterclockwise around point $A$ until $A P$ coincides with $A B$. This process is then continued by rotating the triangle around points $B$, $C$, and $D$ respectively, until point $P$ returns to its original position. The length of the path traced by point $P$ from the start to the end is ( ).
( A) $\frac{7}{2} \pi a$
(B) $\frac{13}{4} \pi a$
(C) $\frac{19}{6} \pi a$
(D) $\frac{25}{8} \pi a$
6. C.

As shown in Figure 5, the path that point $P$ travels is composed of three circular arcs with a radius of $a$ and central angles of $210^{\circ}$, $210^{\circ}$, and $150^{\circ}$. Therefore, the total length is
$$
2 \pi a\left(\frac{210^{\circ}}{360^{\circ}} \times 2+\frac{150^{\circ}}{360^{\circ}}\right)=\frac{19}{6} \pi a .
$$