  A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in
which vertices are mapped to grid points, edges are mapped to horizontal and
vertical straight-line segments, and faces are drawn as rectangles. Sometimes
this latter constraint is relaxed for the outer face. In this paper, we study
rectangular drawings in which the edges have unit length. We show a complexity
dichotomy for the problem of deciding the existence of a unit-length
rectangular drawing, depending on whether the outer face must also be drawn as
a rectangle or not. Specifically, we prove that the problem is NP-complete for
biconnected graphs when the drawing of the outer face is not required to be a
rectangle, even if the sought drawing must respect a given planar embedding,
whereas it is polynomial-time solvable, both in the fixed and the variable
embedding settings, if the outer face is required to be drawn as a rectangle.
