  We aim to constrain the geometrical extent of the chromosphere of non-binary
K giant stars and detect any spatial structures in the chromosphere. We
performed observations with the CHARA interferometer and the VEGA beam combiner
at optical wavelengths. We observed seven non-binary K giant stars. We measured
the ratio of the radii of the photosphere to the chromosphere using the
interferometric measurements in the Halpha and the Ca II infrared triplet line
cores. For beta Ceti, spectro-interferometric observations are compared to an
non-local thermal equilibrium (NLTE) semi-empirical model atmosphere including
a chromosphere. The NLTE computations provide line intensities and contribution
functions that indicate the relative locations where the line cores are formed
and can constrain the size of the limb-darkened disk of the stars with
chromospheres. We measured the angular diameter of seven K giant stars and
deduced their fundamental parameters: effective temperatures, radii,
luminosities, and masses. We determined the geometrical extent of the
chromosphere for four giant stars. The chromosphere extents obtained range
between 16% to 47% of the stellar radius. The NLTE computations confirm that
the Ca II/849 nm line core is deeper in the chromosphere of ? Cet than either
of the Ca II/854 nm and Ca II/866 nm line cores. We present a modified version
of a semi-empirical model atmosphere derived by fitting the Ca II triplet line
cores of this star. In four of our targets, we also detect the signature of a
differential signal showing the presence of asymmetries in the chromospheres.
Conclusions. It is the first time that geometrical extents and structure in the
chromospheres of non-binary K giant stars are determined by interferometry.
These observations provide strong constrains on stellar atmosphere models.
