New calculations for the total reaction cross sections (${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$) are carried out within the framework of nuclear transport theory combined with the optical limit of the Glauber theory. The average nucleon-nucleon collision number is obtained based on the nuclear transport theory. The probability of nucleon-nucleon collision, which is assumed to be a Possion distribution, is calculated, and then ${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$ is calculated by using the optical limit of the Glauber theory. The calculated ${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$ are in agreement with the experimental data for reactions induced by both the stable and the exotic nuclei. In addition, the effects of some ingredients on ${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$ can be easily analyzed with the present method. Among them the effects are mainly from Pauli blocking, the nuclear equation of state, the in-medium nucleon-nucleon cross section ${\mathrm{\ensuremath{\sigma}}}_{\mathit{N}\mathit{N}}$, the effective nucleon density distribution, and the dynamics of nucleus-nucleus collisions. From the calculated energy dependence of ${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$, it is found that the total reaction cross sections are sensitive to the nuclear equation of state when the bombarding energy is below 100 MeV/nucleon and also sensitive to the in-medium nucleon-nucleon cross section in the full researched energy range. Provided that the same nuclear equation of state is applicable to the reactions induced by both the stable and the exotic nuclei in the lower energy range, we seem to have that the in-medium nucleon-nucleon cross section in the exotic nuclei may be larger than in the stable nuclei.