The specific area of the solid-liquid interface of an assembly of dendrites is an important integral measure of the morphology of the microstructure forming during alloy solidification. It represents the inverse of a characteristic length scale and is needed for the prediction of solidification defects and material properties. In the present study, the evolution of the interfacial area of dendrites is analysed using 3D phase-field simulations. A general evolution equation is developed for the specific interface area as a function of time and solid volume fraction that accounts for the effects of growth, curvature-driven coarsening and interface coalescence. The relation is validated using data from previously performed synchrotron X-ray tomography and isothermal coarsening experiments. It is found to be valid for arbitrary and even varying cooling rates and for a wide range of binary alloys. The rate constant in the evolution equation is successfully related to alloy properties.