The "obvious" solution that I think you're describing is the Hammersley sofa (1968), which has A=2.2074.
For a long time, it was thought this might be the optimal shape, but it was never proven. And it couldn't have been because it turns out that you can do better: the Gerver sofa (1992) is a more complicated shape, composed of 18 curve segments and has A=2.2195.
Nobody knew whether there might be an even better shape until now (assuming the proof holds up).