We report the observations in the K band of the red supergiant star α Orionis and of the bright giant star α 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 μm. The full set of data can be explained if a 2055 K layer with optical depths , and is added 0.33  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 [CITE] when interpreting the spectra of strong molecular lines.