The rock and rock-ice mixtures of the spherical shells comprising terrestrial body interiors have thermally determined viscosities well described by an Arrhenius law; however, the implied viscosity contrasts of the activation energies applicable to such bodies exceed a billion orders of magnitude. Accordingly, the numerical modelling of such systems is subject to challenges arising from high viscosity gradients. Here, we investigate the influence of core radius, surface temperature, and convective vigour by investigating their respective impact on thermal structure in two-dimensional spherical annuli. To model the behaviour of a fluid characterized by a viscosity with an Arrhenius temperature dependence, we implement a maximum cut-off viscosity to limit viscosity magnitude in cold regions. We first demonstrate the importance of the choice of a cut-off viscosity, in particular, in response to the curvature factor, defined as the core radius normalized by the outer shell radius (f=Rcmb/Rsurf). As a general result, we find the mean temperature decreases with core size, and that a viscosity contrast of 10^7 results in a stagnant-lid surface across the cases studied(0.3<=f<=0.7). Inverting the results from over 80 stagnant-lid models (from a total of approximately 180 calculations), we apply an energy balance model for heat flow across the thermal boundary layers to obtain a predictive equation for the mean temperatures in the convective region. Our findings can be applied to calculating the internal temperature of cooling spherical shells when core radius and surface temperature are specified, with implications for parameterized thermal history models for stagnant surface planets and moons.