There are some potential uses for round LCDs, for example smart watches. Searching reveals couple news from while ago but I haven't seen any in real use. What is the main reason why these non rectangular displays are not common?
http://www.ubergizmo.com/2008/05/lg-circular-shaped-lcd-display/
Best Answer
It is very difficult to cut round shapes out of glass or produce concave corners. Straight cuts which go all the way across a reasonably-thin sheet of non-tempered glass can be made quickly and reliably by scoring a line and trying to flex the glass there. It's not necessary to score very deep, nor very thoroughly--all that's necessary is to ensure that the scored part breaks before anything else, and the failure of the scored part will tend to cause the entire sheet to fail on that line. Making a curve is much more difficult. If one scores a curve in glass and tries to break it along the curve, the fracture will start somewhere on the score, and depending upon the curvature and depth of the score, the crack may follow the score for some distance, but the scoring must be very deep to have much useful effect, and even when the scoring is deep the crack will still often diverging from it. If one is trying to form an outside curve, one may with some effort be able to to snap off the places where the glass fails to break cleanly on the score, but the amount of work required will be much greater than for a straight break, and the results will be much rougher. Someone who's hand-making stained or fused glass as a hobby project might not mind the amount of labor required to cut curved pieces, or mind the fact that sometimes breaks won't be confined to the parts of the glass one is trying to remove (if one does large pieces first, one may be able to use broken "large pieces" as source material for smaller ones). Such issues, however, would render score-and-cut an ineffective method for mass production.
The only way to effectively mass-produce round displays would be to saw them or machine them. That's much slower than the score-and-cut process used for rectangular displays. It's certainly possible to mass-produce cut glass circles, but doing it with LCDs (where the cut circles would have to line up with the printed patterns on the LCDs) would be harder than doing it with featureless sheets of glass.
Producing octagon-shaped LCDs would probably not be overly expensive compared with square ones--start by scoring a square grid, then add a score to each corner; the leverage available to snap off the corners would make it fairly easy. A 16-sided shape might be possible (start by forming an octagon, then snap off a little bit off each of the 8 corners, but much less leverage would be available on the corners, making it harder to snap them cleanly. Unless the product's physical dimensions would absolutely require the use of a round display, going with a rectangular display would be cheapest, and I would expect an octagonal display would probably be second-cheapest. Its ratio of maximum radius to minimum radius wouldn't be as good as a circle (whose ratio would be 1:1 of course) but at about 1.088 it would be much better than that of a square (whose ratio would be 1.414).