We report the observations in the K band of the red supergiant star \alpha  Orionis and of the bright giant star \alpha  Herculis with the FLUOR beamcombiner at the IOTA interferometer .
The high quality of the data allows us to estimate limb-darkening and derive precise diameters in the K band which combined with bolometric fluxes yield effective temperatures .
In the case of Betelgeuse , data collected at high spatial frequency although sparse are compatible with circular symmetry and there is no clear evidence for departure from circular symmetry .
We have combined the K band data with interferometric measurements in the L band and at 11.15 \mu m. The full set of data can be explained if a 2055 K layer with optical depths \tau _ { K } = 0.060 \pm 0.003 , \tau _ { L } = 0.026 \pm 0.002 and \tau _ { 11.15 \mu m } = 2.33 \pm 0.23 is added 0.33 R _ { \star } above the photosphere providing a first consistent view of the star in this range of wavelengths .
This layer provides a consistent explanation for at least three otherwise puzzling observations : the wavelength variation of apparent diameter , the dramatic difference in limb darkening between the two supergiant stars , and the previously noted reduced effective temperature of supergiants with respect to giants of the same spectral type .
Each of these may be simply understood as an artifact due to not accounting for the presence of the upper layer in the data analysis .
This consistent picture can be considered strong support for the presence of a sphere of warm water vapor , proposed by [ ] when interpreting the spectra of strong molecular lines .