Whispering-gallery (WG) modes in photonic microdevices made of dielectric circularly planar resonators are analyzed. The wave equation is solved by using the method of separation of variables based on the eigenvalue technique. The resonant frequency at an azimuthal mode is decided by iteration using the bisection method based on the continuity conditions at the resonator peripheral boundary. The radial mode is determined by the critical points of the field intensity profile in the radial direction via the first derivative test. The resonance frequencies as well as the E-field distributions in two exemplary small resonators are presented for a variety of modes. Comparison with numerical predictions is conducted, and a good agreement is found. The geometric optics method is found inappropriate for small resonators.