  We deal with the relaxed area functional in the strict $BV$-convergence of
non-smooth maps defined in domains of generic dimension and taking values into
the unit circle. In case of Sobolev maps, a complete explicit formula is
obtained. Our proof is based on tools from Geometric Measure Theory and
Cartesian currents. We then discuss the possible extension to the wider class
of maps with bounded variation. Finally, we show a counterexample to the
locality property in case of both dimension and codimension larger than two.
