  The variation in mean-square nuclear charge radius of xenon isotopes was
measured utilizing a new method based on extreme ultraviolet spectroscopy of
highly charged Na-like ions. The isotope shift of the Na-like D1 (3s
$^{2}$S$_{1/2}$ - 3p $^2$P$_{1/2}$) transition between the $^{124}$Xe and
$^{136}$Xe isotopes was experimentally determined using the electron beam ion
trap facility at the National Institute of Standards and Technology. The mass
shift and the field shift coefficients were calculated with enhanced precision
by relativistic many-body perturbation theory and multi-configuration
Dirac-Hartree-Fock method. The mean-square nuclear charge radius difference was
found to be $\delta<r^2>^{136, 124}$ = 0.269(0.042) fm$^2$. Our result has
smaller uncertainty than previous experimental results and agrees with the
recommended value by Angeli and Marinova [I. Angeli and K. P. Marinova, At.
Data and Nucl. Data Tables {\bf 99}, 69-95 (2013)].
