Symplectic capacities are measurements of symplectic size. They are often given as lengths of closed orbits of a certain canonical vector field, and so connect embedding problems in symplectic geometry with dynamics. I will start by introducing recent joint work showing how to recover the volume of a natural class of symplectic 4-manifolds from a family of symplectic capacities, called ECH capacities. I will then explain several applications of this formula to dynamics, for example to studying surface diffeomorphisms and generalizations of the three-dimensional Weinstein conjecture. I will close by briefly discussing some possible first steps for finding analogous formulas in other dimensions.