The Baade-Wesselink method, also known as the pulsation parallax method allows us to estimate distances to individual pulsating stars. Accurate geometric parallaxes obtained by the Gaia mission serve us in the calibration of the method and in the determination of its precision. The method also provides a way of determining mean radii of pulsating stars. The main aim of this work is to determine the scatter and possible dependence of p-factors of RR Lyrae stars on their pulsation periods. The secondary objective is to determine mean radius - period relations for these stars. Our calibrations for RR Lyrae stars are based on photometric data gathered at the Cerro Murphy Observatory and parallaxes from the Data Release 3 of the Gaia space mission. We obtained spectroscopic data specifically for this project using high resolution spectrographs. We use the Infrared Surface Brightness (IRSB) version of the method that relies on a surface brightness - color relation dependent on the (V-K) color. It allows us to estimate stellar angular diameters while variations of the stellar radius are being traced using measurements of the stellar radial velocity obtained from spectroscopy. We present results based on four different empirical surface brightness-color relations - three of them being relations for dwarfs and subgiants and one for classical Cepheids. We present our calibration of projection factors and determination of the mean radii for nine Galactic RR Lyrae stars. We obtain the spread of p-factors of around 0.07-0.08 for our sample of RR Lyrae stars from the solar neighborhood. However, depending on a SBCR, we also find relations between the p-factor and the pulsation period for RRab stars with the rms scatter around the relation of around 0.05, but with relatively large uncertainty of relations' parameters. We present relations between the mean radius and period for RR Lyrae pulsating in the fundamental mode with the rms scatter around the relation of 0.012 solar radii . We observe a clear offset between p-factors obtained using the IRSB technique (with mean p between 1.39 and 1.45) and values inferred using the SPIPS tool. It confirms that different implementations of the Baade-Wesselink method are sensitive to various components of the p- factor. On the other hand, we obtain a similar scatter of p of as observed in a previous study based on the SPIPS tool. Our period-radius relations are in a good agreement with both the inference based on SPIPS and theoretical predictions.