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The thickness and surface area of cortex are genetically distinct aspects of brain structure, and may be affected differently by age. However, their potential to differentially predict age and cognitive abilities has been largely overlooked, likely because they are typically aggregated into the commonly used measure of volume. In a large sample of healthy adults (N=647, aged 18-88), we investigated the brain-age and brain-cognition relationships of thickness, surface area, and volume, plus five additional morphological shape metrics. Cortical thickness was the metric most strongly associated with age cross-sectionally, as well as exhibiting the steepest longitudinal change over time (subsample N=261, aged 25-84). In contrast, surface area was the best single predictor of age-residualized cognitive abilities (fluid intelligence), and changes in surface area were most strongly associated with cognitive change over time. These findings were replicated in an independent dataset (N=1345, aged 18-93). Our results suggest that cortical thickness and surface area make complementary contributions the age-brain-cognition triangle, and highlight the importance of considering these volumetric components separately.