The interaction cross sections (σI) of the very neutron-rich carbon isotopes 19C, 20C and 22C have been measured on a carbon target at 307, 280, and 235 MeV/nucleon, respectively. A σI of 1.280±0.023 b was obtained for 22C, significantly larger than for 19,20C, supporting the halo character of 22C. A 22C root-mean-squared matter radius of 3.44±0.08 fm was deduced using a four-body Glauber reaction model. This value is smaller than an earlier estimate (of 5.4±0.9 fm) derived from a σI measurement on a hydrogen target at 40 MeV/nucleon. These new, higher-precision σI data provide stronger constraints for assessing the consistency of theories describing weakly bound nuclei.