Many features of the behaviour of mantle convection in the terrestrial planets are dependent on the relative sizes of the core and planet surface areas. In previous studies implementing Cartesian geometry models, convection in strongly temperature-dependent viscosity fluids was investigated for a range of viscosity contrasts and Rayleigh numbers. The viscosity contrasts give rise to convective regimes widely described as mobile-lid, transitional (or sluggish)-lid, and stagnant-lid. For systems with small core-to-planet radii ratios, f, the combinations of Rayleigh number and viscosity contrast required for stagnant-lid convection differ substantially from what is found for f values greater than 0.5. Stagnant-lid convection is achieved in basal heated Cartesian boxes employing non-dimensional viscosity contrasts larger than 10^4 and 10^5 in basal heated spheres (ex: Earth-like f=0.5). Similar contrasts in small core planets instead produces weak convection (low vertical heat advection in the interior). Because full 3D shell calculations are computationally costly, 2D spherical annulus convection was employed to test for stagnant-lid behaviour at extreme viscosity contrasts (up to 13 orders of magnitude). The main features of stagnant-lid convection in ​small core planets initiated from a perturbed conductive temperature profile include an extremely weak fluid with bottom Rayleigh number ecceeding 10^10 at the core-mantle-boundary, thin conducting (stagnant)-lid, and a hot viscous convecting layer separated by vigourous upwellings.