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We focus on empirically measure the p-factor of a homogeneous sample of 29LMC and 10 SMC Cepheids for which an accurate average LMC/SMC distance wereestimated from eclipsing binary systems. We used the SPIPS algorithm, which isan implementation of the BW method. As opposed to other conventional use, SPIPScombines all observables, i.e. radial velocities, multi-band photometry andinterferometry into a consistent physical modeling to estimate the parametersof the stars. The large number and their redundancy insure its robustness andimproves the statistical precision. We successfully estimated the p-factor ofseveral MC Cepheids. Combined with our previous Galactic results, we find thefollowing P-p relation: -0.08(log P-1.18)+1.24. We find no evidence of ametallicity dependent p-factor. We also derive a new calibration of the P-Rrelation, logR=0.684(log P-0.517)+1.489, with an intrinsic dispersion of 0.020.We detect an IR excess for all stars at 3.6 and 4.5um, which might be thesignature of circumstellar dust. We measure a mean offset of $\Deltam_{3.6}=0.057$mag and $\Delta m_{4.5}=0.065$mag. We provide a new P-p relationbased on a multi-wavelengths fit, and can be used for the distance scalecalibration from the BW method. The dispersion is due to the MCs width we tookinto account because individual Cepheids distances are unknown. The new P-Rrelation has a small intrinsic dispersion, i.e. 4.5% in radius. Such precisionwill allow us to accurately apply the BW method to nearby galaxies. Finally,the IR excesses we detect raise again the issue on using mid-IR wavelengths toderive P-L relation and calibrate the $H_0$. These IR excesses might be thesignature of circumstellar dust, and are never taken into account when applyingthe BW method at those wavelengths. Our measured offsets may give an averagebias of 2.8% on the distances derived through mid-IR P-L relations.

Observational calibration of the projection factor of Cepheids IV. Period-projection factor relation of Galactic and Magellanic Cloud Cepheids