Open dielectric waveguides are structures used to guide electromagnetic energy in integrated circuits above the cutoff or as leaky wave antennas propagating the energy transversely out of the waveguide in a narrow region below the cutoff. In this study, the related operating regions for the hybrid EH modes of a cylindrical dielectric rod were obtained analytically. Analyses of the leaky wave characteristics of the hybrid EH modes for various radii of the rod and various dielectric constant values were performed. The guided modes existing above the cutoff with a pure real propagation constant, and the leaky wave modes existing below the cutoff with a complex propagation constant, were obtained from the coefficient matrix of the characteristic equations system of the structure using the bisection method and Davidenko's method, respectively. Additionally, the guided modes of the structure were obtained and designated in the light of previous studies in the literature. The results show that the frequency spectrum of the antenna mode region increases as the value of the dielectric constant and the radius of the dielectric rod decrease. In addition, a circular dielectric with a smaller radius and dielectric constant had a larger frequency spectrum in the leaky wave antenna applications.