  Let $\mathbf{P}_{2n+2}$ be the regular polygon with $2n+2$ vertices, and let
$\theta$ be the rotation of 180$^\circ$. Fomin and Zelevinsky proved that
$\theta$-invariant triangulations of $\mathbf{P}_{2n+2}$ are in bijection with
the clusters of cluster algebras of type $B_n$ or $C_n$. Furthermore, cluster
variables correspond to the orbits of the action of $\theta$ on the diagonals
of $\mathbf{P}_{2n+2}$. In this paper, we associate a labeled modified snake
graph $\mathcal{G}_{ab}$ to each $\theta$-orbit $[a,b]$, and we get the cluster
variables of type $B_n$ and $C_n$ which correspond to $[a,b]$ as perfect
matching Laurent polynomials of $\mathcal{G}_{ab}$. This extends the work of
Musiker for cluster algebras of type B and C to every seed.
