It has been known for some time that the slope of the species–area relationship increases asymptotically at broad spatial scales when richness is plotted against area on logarithmic axes. At continental to global scales, species–area relationships are determined to a large extent by the abundance and size distribution of species ranges. Here we present an analytical model that explicitly quantifies the effects of range size on species–area relationships. The model shows how range size and plot area interact to control the form of species–area relationships at broad spatial scales. It also demonstrates how changes in spatial scale affect biodiversity patterns by changing the relative influence of range size and range abundance on species richness. Our model provides an explanation for the broad-scale upturn of the species–area relationship, but more work is needed to incorporate the effects of range size, habitat heterogeneity, individual sampling and other variables into a unified framework that can account for species–area relationships at all scales.