The total cross sections for the 6, 8He + 28Si and 9Li + 28Si reactions are measured in the beam energy range of 12–39 A MeV via transmission method with registration of prompt neutron and gamma radiation by a spectrometer with six scintillation detectors. The technique taking into account the experimental values of the efficiency of registering gamma radiation with different multiplicities and the number of triggered spectrometer detectors is used to analyze the experimental data. The experimental data are compared to data obtained earlier in other studies. The values of the total cross section for 8He + 28Si reaction at energy around 20 A MeV are close to those of the total cross section for the 6He + 28Si reaction. The total cross section for the 8He + 28Si reaction is appreciably higher than the total cross section for the 6He + 28Si reaction in the range of 12–16 A MeV.