ophisticated solvers will see at once that, with the same amount of patty sticking out, the buns are...

 ...the same size.

Not much challenge there.  Oh, but wait.  Apart from product differentiation or other marketing considerations, maybe there is an economic advantage of one hamburger design over the other.  What might that be?

The square bun may be volumetrically more efficient for shipping and storage than the round bun; however, that advantage will surely be offset by complexities in baking and packaging.  Besides, the square patty enjoys those same advantages of squareness while requiring less space for refrigeration.  Ah, there's a clue. {Hypernote}

onsider the difference in surface area for the two types of hamburger patties.  Viewed from above, of course, the surface areas are the same -- a circle of diameter d having an area equal to a square with side s, such that s =1/2 d / 2.  However, the surface area around the perimeter of the patty is  d T for the circular patty, where T represents the thickness,  and 4 s T = 21/2 d T for the square patty, a difference of about 13%.

If we assume that the thickness of the traditional patty T is, say, 1/10th the diameter d, then it is easy to show that the surface-area-to-volume ratio of the circular patty is 5% smaller than that of the square patty.  That means the square patty would cook about 5% faster

Fast food, remember?

In the Battle of the Buns, preparation promptness of peculiar patties preponderates planform preferences, presumably.

Hypernote

If you're old enough, you may recall the days when dairy products were delivered to your door.  In returnable bottles.  That form of package has long ago given way to containers that maximize volumetric efficiency in refrigeration.

In geometrical shape deployed in the supermarket has the general name parallelepiped (now usually pronounced , P@ER&L&L&PAEP&D in recognition of its etymology in Greek and meaning a body "having parallel planes").  In general a parallelepiped  is a three-dimensional figure formed by six parallelograms.

Three equivalent definitions of parallelepiped are...

• prism of which the base is a parallelogram,
• hexahedron of which each face is a parallelogram, and
• hexahedron with three pairs of parallel faces.
• Parallelepipeds are a subclass of the prismatoids. The cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped.  So then, what is the proper geometrical name for the milk carton?