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This paper reports on a detailed experimental study of quasioptical dielectric resonators (QDRs) designed in the form of a truncated cone and excited on whispering-gallery modes. One aim of this paper is to study of the eigenfrequency spectrum and quality factors of single-crystal sapphire resonators. A second aim is to show the applicability of the resonator technique not only in optical but also in the millimeter-wave range, in particular, for studying the surface resistance (Rs) of high-temperature superconducting films. The measured frequencies of QDR as a function of the wedge angle of the cone resonator are in a good agreement with those frequencies simulated using the CST Microwave Studio 2006 program. Additionally, electrodynamic characteristics of the cone resonator and hemispherical dielectric resonator with conducting endplate are compared.