For any vector $r = (r_1,...,r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in $\mathbb{R}^3$ with n-sides with lengths $r_1,..., r_n$. $M_r$ can also be described as the symplectic reduction for the natural action of the torus $U^n_1$ of diagonal matrices in the unitary group $U_n$, on the complex Grassmannian of $2$-planes $Gr_2(\mathbb{C}^n)$ (Hausmann-Knutson). An interesting application of the volume formula for $M_r$ (Vu The Khoi; ---) is the calculation of the cohomology ring $H^*(M_r)$. In fact we show that the coefficients of the Duistermaat-Heckman polynomial encode all the necessary information on the generators and relators of $H^*(M_r)$. The ring $H^*(M_r)$ has already been determined by Hausmann and Knutson, but the technique that we present involves a thorough analysis of how the diffeotype of $M_r$ changes as $r$ crosses a wall in $\mu_{U^n_1}(Gr_{2,n})$ (which we believe has an independent interest) and perhaps gives a geometrically more direct comprehension of $H^*(M_r)$.