Empirical field data describing daily and seasonal cycles in body temperature (T_b) of free-ranging Crocodylus porosus (32–1010 kg) can be predicted by a mathematical analysis. The analysis provides a mechanistic explanation for the decreased amplitude of daily cycles in T_b and the increase in ‘average’ T_b with increasing mass. Assessments of ‘average’ daily T_b were made by dividing the integral of the difference between measured values of T_b and minimum operative temperature by the period of integration, to yield a thermal index expressing relative ‘warmth’ of crocodiles. The average daily T_b of a 1010 kg crocodile was 3.7 °C warmer than that of a 42 kg individual in summer and 1.9 °C warmer than that of a 32 kg individual in winter. The success of this mathematical approach confirms that crocodiles are simple ectotherms and that there is unlikely to be a significant contribution to their thermal biology from physiological mechanisms. Behaviour, however, is very important even in large individuals. Crocodiles in the field typically move daily between land and water in cycles that vary seasonally. We predicted T_b for the reverse of these behavioural cycles, which more than doubled seasonal fluctuations in T_b compared with the observed fluctuations. We were also able to predict the T_b of very large, dinosaur-sized crocodiles in a similar climate to that at our study site. A 10 000 kg ‘crocodile’, for example, would be expected to have a T_b of 31 °C in winter, varying by less than 0.1 °C during a day when operative temperatures varied by nearly 20 °C, from 20 to 38 °C. The study confirms that, in low latitudes at least, large dinosaurs must have had an essentially high and stable value of T_b, without any need for endothermy. Also, access to shade or water must have been crucial for the survival of large dinosaurs at low latitudes. Furthermore, the finding of increasing ‘average’ T_b as ectotherms grow larger may have implications for the metabolic rates of very large reptiles, because the Q₁₀ effect could counteract the downscaling of metabolic rate with mass, an effect that seems not to have been recognised previously.