13. Given a regular tetrahedron $P-ABC$ with a volume of $9 \sqrt{3}$, the dihedral angle between the side face $PAB$ and the base $ABC$ is $60^{\circ}$. Point $D$ is a point on line segment $AB$ such that $AD=\frac{1}{6} AB$, point $E$ is a point on line segment $AC$ such that $AE=\frac{1}{6} AC$, and point $F$ is the midpoint of line segment $PC$. Plane $DEF$ intersects line segment $PB$ at point $G$. Find the area of quadrilateral $DEFG$.
Given a regular tetrahedron $P-ABC$ with the base edge length $a$ and height $h=\frac{1}{2} a$, its volume is
$$
V=\frac{1}{3} \cdot \frac{\sqrt{3}}{4} a^{2} \cdot \frac{1}{2} a=\frac{\sqrt{3}}{24} a^{3}=9 \sqrt{3} \Rightarrow a=6 .
$$

As shown in the figure, $BC \parallel ED \Rightarrow BC \parallel FG \Rightarrow ED=1$,
$FG=3, MN=\frac{5}{6} AM=\frac{5 \sqrt{3}}{2}$.
Also, $PM=2 \sqrt{3} \Rightarrow QM=\sqrt{3}$,
In $\triangle QMN$,
$$
NQ^{2}=\frac{75}{4}+3-\frac{5 \sqrt{3}}{2} \cdot \sqrt{3}=\frac{57}{4} \Rightarrow NQ=\frac{\sqrt{57}}{2} \Rightarrow S_{DEFG}=\sqrt{57}
$$