6. If the edge length of a cube is $a$, then the area of the largest section perpendicular to the cube's diagonal is ( ).
(A) $\frac{3 \sqrt{2}}{4} a^{2}$
(B) $\sqrt{2} a^{2}$
(C) $\frac{\sqrt{3}}{2} a^{2}$
(D) $\frac{3 \sqrt{3}}{4} a^{2}$
6. D.

The maximum cross-section through the center of the cube is a regular hexagon with a side length of $\frac{\sqrt{2}}{2} a$, and the area is
$$
\frac{\sqrt{3}}{4} \times \frac{1}{2} a^{2} \times 6=\frac{3 \sqrt{3}}{4} a^{2} .
$$