During the chirality analysis of various molecules containing rings, we assume for convenience that the rings are completely flat/planar and if the molecule has a plane of symmetry, it is achiral. However, does the fact that say, 6-membered rings actually exist in a chair or boat conformation have any effect on our final answer (it has a plane of symmetry if it is completely planar, but does not if it is in its 3D conformation)?
I'm a little confused because the odd, puckered conformation of various rings don't seem like they would have a plane of symmetry. Take for example, the molecule below. In the 2D drawing, we see 1 plane of symmetry, it goes through $\ce {Br}$ and cuts through the ring in a perpendicular fashion. However, in the 3D drawing, the plane we identified earlier doesn't seem like a plane of symmetry anymore.
Of course, in the 2D picture, the molecules would seem to be equivalent no matter if the bond were wedged or dashed, but why do they appear to be different if they are drawn in 3D?
