Meririam-Webster defines facet as a small plane surface (as on a cut gem). Traditional flat faceting is the systematic cutting and polishing of that flat plane surface in a geometric pattern on transparent gem material. . The facet angle and order of the facets is determined by the angle and rotation of the arm in relationship to a “turntable”. The facets are cut (a misnomer, they are really ground) utilizing diamond-charged, flat disks (laps) that rotate horizontally on a platter much like a record on a turntable. The gem is held on an “arm” and this arm is controlled both horizontally and radially.
Traditional flat faceting machine
This view (above) of an amethyst being faceted gives a good view of the lap (platter) that is charged with diamond.
This is a close up view (above) showing gauges that help regulate the angle of each facet. The larger gauge in the background increases accuracy to very close tolerance.
This view shows the gear that places the facets at specific places around the gem. The combination of the angle gauge and the gear above, when used by a skilled gem designer can create an infinite number of shapes of gems and placement of facets.
Flat facets are shaped (kite shape, triangle shape, rectangle shape, etc) by their relationship to their neighboring facets and the angles at which they intersect one another. Flat facets are two dimensional - they have only length and width. This two dimensional effect causes the facets to be able to reflect or not reflect depending on the angle of the light in relationship to the facets. If the light hits inside the angle of incidence, the light will go right through the facet with no return to the surface and therefore to the eye. Since one cannot determine the angle that the light will enter the gem, light leakage through the bottom of the gem is a major problem. The goal of most faceting is to create total internal reflection and flat facets make this impossible because one cannot absolutely control the angle of entry of the light. In fact, they maximize the possibility of light leakage based upon the random nature of the angle that the light will enter the gem.

As you can see from the diagram above that a round shape is actually formed by a series of flat facets.