Perhaps in the professional pizza box making industry you have some fancy technical terminology and a secret handshake, but in colloquial English “corrugated” is an adjective, and the stuff pizza boxes are made from is called corrugated cardboard. Or, in situations where the particular type of cardboard is not relevant to what’s being discussed, we might just call it “cardboard” for short. Sheer laziness, but that’s amateurs for you.
Interesting, thanks for posting. I wonder if it’s the sort of thing where if more people built them, the money would justify innovation, troubleshooting, etc. It looks like the sort of thing they might build on another planet some day.
I guess she’s supposed to be looking pensive, but I think the lady in that commercial looks like she’s been kicked in the head by a horse.
Plus, the cardboard comes in flat sheets. There is no way to make the vertical sides curved without kinking the corrugations making it somewhat weaker. Domino’s boxes with a square lid enclosing an octagonal interior is probably the best approximation.
In 1962 or so an Italian bakery near my Hebrew School sold Sicilian pizza which was baked in big rectangles. Damn good pizza too, though my teachers were not thrilled with the sausage.
Even worse - when the pizza delivery guy drives to your house on a parkway, and then parks in your driveway. 
Surely there are uses for square pieces of cardboard with a round hole cut out—packing material for large pots, platters, and other crockery, perhaps?
it’s easier to make pizzas into circles
it’s easier to make boxes into squares.
Same reason why hamburgers are made circular
and put into square boxes.
Also squares/boxes are stackable where as circular is more difficult to stack.
That’s different. The area of the burger is intended to be such that when placed on a round bun, the four corners stick out. Makes the customer think it’s an oversized patty. But obviously you can’t have part of a pizza sticking out of the box.
Pizza is round, because that shape is the more natural shape for formed dough. It also is easier to cook consistently that a square pizza.
When cooked individually in a traditional pizza oven, a square pizza will get burned corners.
Square is possible, but typically require a fancier oven with conveyor belt and for the pizza to be cut into squares after cooking, thus losing that essential part of Pizza that is The Crust.
Thus, almost all Pizza are circular(ish)
Boxes tend to be square(well, rectangular), because that are manufactured as flat sheets of stuff that is then folded into a container.
And if you want a container with an attached lid, that lid’s hinge HAS to be on a straight edge.
It is possible to fold a flat sheet into a complex shape, but this requires either more cut-out wastage or much-much more complex folding. (think Origami)
Thus, almost all Pizza boxes are rectangular.
The best fit rectangle for a single circular object is… a square.
p.s.
I have seen rectangular pizza boxes… typically for a 2-pizza special, or for a pizza-with-side-dish combo.
I have also seen octagonal pizza boxes. They tended to break along the hinge line.
I have also seen a pizza box shaped like a Clam. (round body, flat hinge along one edge).
it was heck to carry without the thing flopping open unexpectedly!
The round pizza boxes I’ve seen are like the Apple version referenced above – pressed paper pulp, not cut from square corrugated board. The one’s I’ve seen in use were the rough pressed paper pulp, not smooth surface like the Apple version.
Everyone knows that pie are round; corn bread are square! 
I make corn bread in a cast iron skillet, so corn bread are round, too.
So in answer to the more general question, “why are disc-shaped products like CDs, CD-ROMs, DVDs, or standard pizza pies, packaged in square boxes?” - I can tell you my experience with one case where I owned one that happened to come packaged in a round (disc shaped) box.
It was a Special Limited Edition DVD release of the movie “Total Recall”, that instead of the usual rectangular plastic case for a DVD, came in a round red metal “steelie” box that was made to look like the planet Mars (where most of the action of the movie takes place… Or does it…?).
Putting this round metal case vertically on a bookshelf with my other DVDs was Not Successful. It constantly rolled out and fell onto the floor. I had to store it flat, like a drink coaster.
Now picture LPs and CD-ROMS - they were usually indexed in bookshelf fashion, with the identifying names on the spine. Round enclosures would have this same problem.
Pizza boxes? Those you could plausibly stack flat and on top of each other in columns, but then if a stack ever got tipped over, they’d roll like coins. Much harder to pick up. Plus, with square boxes you have the option of stacking/storing them sideways as well, if for some reason that worked better.
how fast can you make a round box?
Design time_________?
Construct time__________?
How fast can you make a square box?
Design time___________?
Construct time__________?
How fast can you divide a circle equally___________
how accurate can you cut_________
how fast can you cut squares equally_________
how accurate can you cut__________
how square can you toss dough__________
(things they may have considered before choosing circle pie in round box)
I’ve worked at industrial-type assembly line pizza stores (Straw Hat Pizza Palace) and an authentic Italian restaurant where I learned how to toss the pizzas by hand. At the Piccola Italia, when I made pizza dough, each blank was lovingly separated from the mass, lovingly patted into a loaf shape, and left in an oiled wooden drawer to rise.
At the Pizza Palace, a fifty-pound mass of dough (that I had made in a Hobart industrial dough-hook mixer) was sliced into, well, let’s call them 2.5 pound bricks.* A brick would be place in a rolling device that would (eventually) roll the dough into a quarter-inch thick sheet maybe six feet long, and the pizzarristas (feel free to make up your own word to describe the occupation) would use variously-sized cookie cutters to create the blanks. There was a street-facing window in the store so people could watch pizza crusts being perfunctorily tossed, but that was really just a bit of theater.
*they gave the dough some time to rise before they bricked, wrapped, and refrigerated it. We weren’t barbarians.