Reaction cross sections (sigma(R)) for 19C, 20C and the drip-line nucleus 22C on a liquid hydrogen target have been measured at around 40A MeV by a transmission method. A large enhancement of sigma(R) for 22C compared to those for neighboring C isotopes was observed. Using a finite-range Glauber calculation under an optical-limit approximation the rms matter radius of 22C was deduced to be 5.4+/-0.9 fm. It does not follow the systematic behavior of radii in carbon isotopes with N < or = 14, suggesting a neutron halo. It was found by an analysis based on a few-body Glauber calculation that the two-valence neutrons in 22C preferentially occupy the 1s(1/2) orbital.