The optimum cross-sectional area profile of gas-cooled high-temperature superconductor(HTS) current lead is analyzed to have minimum helium boil-off rate. The conventional constant area HTS lead has much higher helium consumption than the optimum HTS lead considered in this study. The optimum HTS lead has variable cross-sectional area to have constant satiety factor. An analytical formula of optimum shape of lead and temperature profile are obtained. For multi-stacking HTS current leads, the optimum tape lengths and minimum heat dissipation rate are also formulated. The developed formulations are applied to the Bi-2223 material, and the differences between constant area, constant safety-factor, and multi-stacking current leads are discussed.