  Let us say that an $n$-sided polygon is semi-regular if it is
circumscriptible and its angles are all equal but possibly one, which is then
larger than the rest. Regular polygons, in particular, are semi-regular. We
prove that semi-regular polygons are spectrally determined in the class of
convex piecewise smooth domains. Specifically, we show that if $\Omega$ is a
convex piecewise smooth planar domain, possibly with straight corners, whose
Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular
polygon $P_n$, then $\Omega$ is congruent to $P_n$.
