A supplement to Ian's answer: Here is the largest-area sofa known, due to Gerver:
![GerverSofa][1]
Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner". Geometriae Dedicata 42 (3): 267–283. (Springer link.)
Added (triggered by @GeraldEdgar's remark). The computational complexity of algorithms grows exponentially in the dimension. Here is an algorithm moving a $4500$-triangle piano through a challenging apartment requiring several tricky maneuvers:
![Piano][2]
Kuffner, James J., and Steven M. LaValle. "RRT-connect: An efficient approach to single-query path planning." Robotics and Automation, 2000. Proceedings. ICRA'00. IEEE International Conference on. Vol. 2. IEEE, 2000.
Not surprisingly, the problem is also called The Piano Mover's Problem.