5. For a regular quadrilateral pyramid, the area ratio of a diagonal section to a side face is $\sqrt{6}: 2$. Then the angle between the side face and the base is ( ).
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{12}$
5. A.

As shown in Figure 3, let the side length of the square base be 1, the height of the pyramid be $h$, and the height of the triangular side be $l$. Then
$$
\begin{array}{c}
A C=\sqrt{2}, \\
\frac{\sqrt{6}}{2}=\frac{\sqrt{2} h}{l} . \\
\text { Hence } \sin \angle P M H=\frac{h}{l} \\
=\frac{\sqrt{3}}{2}, \angle P M H=\frac{\pi}{3} .
\end{array}
$$

Therefore, $\sin \angle P M H=\frac{h}{l}$