11. As shown in Figure 2, in the quadrilateral pyramid $P-ABCD$, the base $ABCD$ is a square with side length $a$, $PD=a$, $PA=PC=\sqrt{2} a$. If a sphere is placed inside this quadrilateral pyramid, then the maximum radius of the sphere is ( ).
(A) $(\sqrt{2}-1) a$
(B) $\sqrt{2} a$
(C) $\left(1-\frac{\sqrt{2}}{2}\right) a$
(D) $a$
11. C.

By the equal volume method, we have
$$
\begin{array}{l}
V_{\text {tetrahedron } P-A B C D}=\frac{1}{3} a^{2} \times a \\
=\frac{1}{3}\left(a^{2}+2 \times \frac{1}{2} a^{2}+2 \times \frac{1}{2} a \times \sqrt{2} a\right) r \\
\Rightarrow r=\left(1-\frac{\sqrt{2}}{2}\right) a .
\end{array}
$$