A scaling framework unifying the markedly different and independently studied cylindrical and spherical shock convergence is presented. For ionizing argon, we show that the focal temperature becomes invariant to shock symmetry and initial shock conditions when scaled by the prefocus shock Mach number, after accounting for pressure effects. The resulting collapsed focal temperature–Mach number relation is governed by the thermodynamics of argon, up to an equilibrium temperature of 35 000 K. Such a scaling enables both predictive estimation of focal temperatures over a wide range of initial conditions and, conversely, determination of the parameters required to achieve a target temperature in a given medium.