1. For a regular triangular prism with base edge length and side edge length both 1, a section is made through one side of the base and the midpoint of the line connecting the centers of the top and bottom bases. What is the area of the section? $\qquad$ .
$=.1 . \frac{4 \sqrt{3}}{9}$.
As shown in Figure 4, let $P$ be the midpoint of the line segment $O_{1} O$ connecting the centers of the upper and lower bases. Draw a plane through $A B$ and $P$ that intersects the extension of $C C_{1}$ at point $E$. Connect $A E$ to intersect $A_{1} C_{1}$ at point $F$, and connect $B E$ to intersect $B_{1} C_{1}$ at point $G$. The trapezoid $A B G F$ is the desired cross-section. Connect $E P$ to intersect $A B$ and $F G$ at points $D$ and $H$, respectively. Since $O$ and $O_{1}$ are the centers of their respective triangles, $D$ is the midpoint of $A B$ and $C D \perp A B$. Also, $C_{1} C \perp A B$, so $A B \perp$ plane $E C D$, making $D H$ the height of the trapezoid.
Given $\frac{E C}{P O}=\frac{C D}{O D}=\frac{3}{1}$, we have $E C=3 P O=\frac{3}{2}$. Therefore, $E C_{1}=\frac{1}{2}$.
Since $\frac{E C_{1}}{E C}=\frac{E H}{E D}=\frac{G F}{A B}$, we get
$G F=\frac{1}{3} A B=\frac{1}{3}$,
$H D=\frac{2}{3} E D=\frac{2}{3} \sqrt{E C^{2}+C D^{2}}=\frac{2 \sqrt{3}}{3}$.
Thus, $S_{\text {moon }}$.