To explore the fundamental properties of superconducting materials used in modern particle accelerators, high precision surface resistance measurements in a dedicated testing equipment is of key importance. The quadrupole resonator, originally developed at CERN, and then successfully modified at the Helmholtz-Zentrum Berlin, is ideally suited for characterization of samples at temperatures of 1.8 K to > 20 K, RF fields of up to 120 mT and frequencies of 433 MHz, 866 MHz and 1.3 GHz. In the past years, this set-up has been subject of intensive research on both its capabilities and limitations. Yet, one of the main challenges is the accuracy of the surface resistance measurement, which is determined by both the uncertainty in the RF measurement and manufacturing imperfections related to the production tolerances such as quenching and chemical polishing processes, etc. In this contribution, we focus on the influence of key geometrical parameters on operating the quadrupole resonator at the third mode, since the surface resistance measurement shows some unexpected behavior for this frequency.