Context :

Aims : We present coordinated near-infrared K -band interferometric and optical spectroscopic observations of the M 1.5 giant $ α $ Cet ( Menkar ) obtained with the instruments VINCI and UVES at the Paranal Observatory .
Spherically symmetric PHOENIX stellar model atmospheres are constrained by comparison to our interferometric and spectroscopic data , and high-precision fundamental parameters of Menkar are obtained .

Methods : Our high-precision VLTI/VINCI observations in the first and second lobes of the visibility function directly probe the model-predicted strength of the limb darkening effect in the K -band and the stellar angular diameter .
The high spectral resolution of UVES of R = 80 000 - 110 000 allows us to confront in detail observed and model-predicted profiles of atomic lines and molecular bands .

Results : We show that our derived PHOENIX model atmosphere for Menkar is consistent with both the measured strength of the limb-darkening in the near-infrared K -band and the profiles of spectral bands around selected atomic lines and TiO bandheads from 370 nm to 1000 nm .
At the detailed level of our high spectral resolution , however , noticeable discrepancies between observed and synthetic spectra exist .
We obtain a high-precision Rosseland angular diameter of \Theta _ { \mathrm { Ross } } =12.20 mas \pm 0.04 mas .
Together with the Hipparcos parallax of 14.82 mas \pm 0.83 mas , it corresponds to a Rosseland radius of R _ { \mathrm { Ross } } =89 \pm 5 R _ { \odot } , and together with the bolometric flux based on available spectrophotometry , to an effective temperature of T _ { \mathrm { eff } } =3795 K \pm 70 K. The luminosity based on these values is L =1460 L _ { \odot } \pm 300 L _ { \odot } .
Relying on stellar evolutionary tracks , these values correspond to a mass M = 2.3 M _ { \odot } \pm 0.2 M _ { \odot } and a surface gravity \log g =0.9 \pm 0.1 ( cgs ) .

Conclusions : Our approach illustrates the power of combining interferometry and high-resolution spectroscopy to constrain and calibrate stellar model atmospheres .
The simultaneous agreement of the model atmosphere with our interferometric and spectroscopic data increases confidence in the reliability of the modelling of this star , while discrepancies at the detailed level of the high resolution spectra can be used to further improve the underlying model .