We prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the Euclidean space with homogeneous Dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves. We consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time. As a consequence of this equivalence, using the well-known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control.