  A nuclear radius of $^{22}$C is investigated with the total reaction cross
sections at medium- to high-incident energies in order to resolve the radius
puzzle in which two recent interaction cross section measurements using $^1$H
and $^{12}$C targets show the quite different radii. The cross sections of
$^{22}$C are calculated consistently for these target nuclei within a reliable
microscopic framework, the Glauber theory. To describe appropriately such a
reaction involving a spatially extended nucleus, the multiple scattering
processes within the Glauber theory are fully taken into account, that is, the
multi-dimensional integration in the Glauber amplitude is evaluated using a
Monte Carlo technique without recourse to the optical-limit approximation. We
discuss the sensitivity of the spatially extended halo tail to the total
reaction cross sections. The root-mean-square matter radius obtained in this
study is consistent with that extracted from the recent cross section
measurement on $^{12}$C target. We show that the simultaneous reproduction of
the two recent measured cross sections is not feasible within this framework.
