  We give a presentation of the cohomology ring of spatial polygon spaces
$M(r)$ with fixed side lengths $r \in \mathbb R^n_+$. These spaces can be
described as the symplectic reduction of the Grassmaniann of 2-planes in
$\mathbb C^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the
torus of diagonal matrices in the unitary group U(n). We prove that the first
Chern classes of the $n$ line bundles associated with the fibration $r$-level
set $\rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \mathbb C).$ By
applying the Duistermaat--Heckman Theorem, we then deduce the relations on
these generators from the piece-wise polynomial function that describes the
volume of $M(r).$ We also give an explicit description of the birational map
between $M(r) $ and $M(r')$ when the lengths vectors $r$ and $r'$ are in
different chambers of the moment polytope. This wall-crossing analysis is the
key step to prove that the Chern classes above are generators of $H^*(M(r))$
(this is well-known when $M(r)$ is toric, and by wall-crossing we prove that it
holds also when $M(r)$ is not toric).
