While connected arithmetic discrete lines are entirely characterized, only partial results exist for the more general case of arithmetic discrete hyperplanes. In the present paper, we focus on the $3$-dimensional case, that is on arithmetic discrete planes. Thanks to arithmetic reductions on a vector $\vect{n}$, we provide algorithms either to determine whether a given arithmetic discrete plane with $\vect{n}$ as normal vector is connected, or to compute the minimal thickness for which an arithmetic discrete plane with normal vector $\vect{n}$ is connected.