Wafers used for making semiconductors are round -- but this wastes quite a few chips around the periphery of the wafer in the fabrication process. Wouldn't it make sense to make the wafer as a square or rectangle instead?
Is there some aspect of the lithography process that requires that the surface be round?
As the wafer material is drawn up out of the molten silicon, it is spun in order to produce a single uniform silicon crystal via the Czochralski process. It is this spinning that produces the round profile of the wafer itself.
The manufacturing process results in a cylinder of silicon. The manufacturers could just square it off and return the trimmings to the pot, but as the whole process evolved to handle round wafers they just don't.
So the direct answer to "Is there some aspect of the lithography process that requires that the surface be round?" is "the machines are designed to accept round wafers, so that's what the wafer makers deliver. The next machines need to work with existing round wafers, so ......" and on it goes.
Some manufacturers will make smaller chips in the corners, it really depends on compatibility and economics. I have heard of foundries specializing in custom imaging elements (which can be quite large) adding smaller imaging devices for "free" around the perimeter. "Free" in this case being far less money than usual.
Even if it was square or rectangular, it wouldn't necessarily make the area usage more efficient -- there is no guarantee that the die size would be a submultiple of the chosen size.
Rectangular wafers would have sharp corners that are more susceptible to stress (which affects uniformity of devices across the wafer) and breakage.
Wafers are spun (rotated at high speed) to uniformly coat photoresist, and to rinse them. Flat edges could induce vibrations as well as cause non-uniform coating.
