Two types of grazing incidence ring resonators for use with free-electron lasers have been investigated. These cavities utilize off-axis conical and flat mirrors and have been designed to operate in the extreme
ultraviolet region of the spectrum. In this paper, a design algorithm that calculates the mirror parameters for propagation of Gaussian TEM00 mode beams in the two cavity types is presented. Results concerning the angular
stability of each type are also shown.