  We computationally study the Fermi arc states in a Dirac semimetal, both in a
semi-infinite slab and in the thin-film limit. We use Cd$_3$A$_2$ as a model
system, and include perturbations that break the $C_4$ symmetry and inversion
symmetry. The surface states are protected by the mirror symmetries present in
the bulk states and thus survive these perturbations. The Fermi arc states
persist down to very thin films, thinner than presently measured
experimentally, but are affected by breaking the symmetry of the Hamiltonian.
Our findings are compatible with experimental observations of transport in
Cd$_3$As$_2$, and also suggest that symmetry-breaking terms that preserve the
Fermi arc states nevertheless can have a profound effect in the thin film
limit.
