Giant stars, and especially C-rich giants, contribute significantly to the chemical enrichment of galaxies. The determination of precise parameters for these stars is a necessary prerequisite for a proper implementation of this evolutionary phase in the models of galaxies. Infrared interferometry opened new horizons in the study of the stellar parameters of giant stars, and provided new important constraints for the atmospheric and evolutionary models.We aim to determine which stellar parameters can be constrained by using infrared interferometry and spectroscopy, in the case of C-stars what is the precision which can be achieved and what are the limitations. For this purpose we obtained new infrared spectra and combined them with unpublished interferometric measurements for five mildly variable carbon-rich asymptotic giant branch stars. The observations were compared with a large grid of hydrostatic model atmospheres and with new isochrones which include the predictions of the thermally pulsing phase. For the very first time we are able to reproduce spectra in the range between 0.9 and 4 $mu$m, and $K$ broad band interferometry with hydrostatic model atmospheres. Temperature, mass, log$(g)$, C/O and a reasonable range for the distance were derived for all the objects of our study. All our targets have at least one combination of best-fitting parameters which lays in the region of the HR-diagram where C-stars are predicted. We confirm that low resolution spectroscopy is not sensitive to the mass and log$(g)$ determination. For hydrostatic objects the $3,mu$m feature is very sensitive to temperature variations therefore it is a very powerful tool for accurate temperature determinations. Interferometry can constrain mass, radius and log$(g)$ but a distance has to be assumed. The large uncertainty in the distance measurements available for C-rich stars remains a major problem.
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