  The M\"obius metric $\delta_G$ is studied in the cases where its domain $G$
is an open sector of the complex plane. We introduce upper and lower bounds for
this metric in terms of the hyperbolic metric and the angle of the sector, and
then use these results to find bounds for the distortion of the M\"obius metric
under quasiregular mappings defined in sector domains. Furthermore, we
numerically study the M\"obius metric and its connection to the hyperbolic
metric in polygon domains.
