  Finite frieze patterns with entries in
$\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$ where $\{p_1,\ldots,p_s\}
\subseteq \mathbb{Z}_{\geq 3}$ and $\lambda_p = 2 \cos(\pi/p)$ were shown to
have a connection to dissected polygons by Holm and Jorgensen. We extend their
work by studying the connection between infinite frieze patterns with such
entries and dissections of annuli and once-punctured discs. We give an
algorithm to determine whether a frieze pattern with entries in
$\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$, finite or infinite, comes
from a dissected surface. We introduce quotient dissections as a realization
for some frieze patterns unrealizable by an ordinary dissection. We also
introduce two combinatorial interpretations for entries of frieze patterns from
dissected surfaces. These interpretations are a generalization of matchings
introduced by Broline, Crowe, and Isaacs for finite frieze patterns over
$\mathbb{Z}$.
