A group G is called cyclic subgroup separable for the cyclic subgroup H if for each $x;{in};G{ackslash}H$, there exists a normal subgroup N of finite index in G such that $x;{ otin};HN$. Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to polygonal products of polycyclic-by-finite groups and free-by-finite groups.