High-temperature effects have a significant impact on the characteristics of aircraft moving at super- and hypersonic speeds. Due to the complexity of setting up a physical experiment, methods of mathematical modelling play an important role in finding the characteristics of hypersonic aircraft. Numerical modelling of super- and hypersonic air flow around the sphere is carried out taking into account high-temperature effects and non-equilibrium chemical reactions. The mathematical model includes equations of gas dynamics, equations of the turbulence model and equations of chemical kinetics. A critical review of the various models that are used to find the stand-off distance is provided. The results of numerical calculations on the distribution of flow quantities in the shock layer and the stand-off distance at different freestream Mach numbers are presented. The results of numerical calculations in a wide range of freestream Mach numbers are compared with the experimental data and computational results of other researchers. Standoff distance is computed at various Mach number and heights during a flight in the atmosphere at hypersonic speed.