6. The number of trapezoids formed by the $4 n+2$ vertices of a regular $4 n+2$-sided polygon is $\qquad$
6. $2 n(n-1)(2 n+1)$ Detailed Explanation: A regular $4 n+2$-sided polygon has $2 n+1$ diameters. All trapezoids can be divided into two categories: one category is parallel to a certain diameter, and there are $(2 n+1)\left(C_{2 n+1}^{2}-n\right)$ such trapezoids; the other category is perpendicular to a certain diameter, and there are $(2 n+1)\left(C_{2 n}^{2}-n\right)$ such trapezoids. Therefore, the total number of trapezoids is $2 n(n-1)(2 n+1)$.