3. In a cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$ with edge length $a$, $X, Y$ are the centers of the squares $A A_{1} B B_{1}$, $B B_{1} C_{1} C$ respectively, and $Z$ is on the diagonal $B D$ such that $D Z=$ $3 Z B$. Then the area of the section through $X, Y, Z$ is $\qquad$
3. The cross-section through $X, Y, Z$ is a rectangle, with the length being the height of the square $a$, and the width being the length of $X Y$, which is $\frac{\sqrt{2} a}{2}$. Therefore, the area of the cross-section through $X, Y, Z$ is $\frac{\sqrt{2} a^{2}}{2}$.