Using the methods of numerical simulation and quantum statistics, based on the thermodynamic quantities of Fermi gas trapped in a general external potential, we studied the thermodynamic properties of a relativistic Fermi gas trapped in hard-sphere potential, gave the analytical expressions of the energy and heat capacity of the relativistic Fermi system, and analyzed the influence of the relativistic effect on the energy and heat capacity. The results showed the energy and heat capacity of the relativistic Fermi system is larger than those of the nonrelativistic case. The larger the relativistic eigenvalue, the lower the change temperature of heat capacity. As the temperature rises, the larger the relativistic eigenvalue, the greater the energy.