The Tetragonoceratidae is a small family of nautilitids

Nautilida

The Nautilida constitute a large and diverse order of generally coiled nautiloid cephalopods that began in the mid Paleozoic and continues to the present with a single family, the Nautilidae which includes two genera, Nautilus and Allonautilus, with six species...

 constituting a part of the superfamily Tainocerataceae

Tainocerataceae

The Tainocerataceae is a superfamily in the cephalopod order Nautilida characterized by straight to loosely coiled shells, generally with a degressed, such that the width is greater than the height, to quadrate whorl section. Many bore spines, ribs, frills, wings, or nodes...

 in which shells are coiled with a generally quadrate whorl section. Coiling is either gyroconic or evoluute with a slight dorsal impression. Flanks diverge from the umbilical to the ventral shoulders so as to make the whorl sections widest close to the venter. Nodes made develop on the flanks and shoulders. The siphuncle is tubular, typically found near the venter. Sutures are with lateral lobes and may possess dorsal and ventral lobes as well.

The Tetragonoceratidae, which is an early offshoot of the Rutoceratidae

Rutoceratidae

The Rutoceratidae are the prototypical nautilids, derived probably from either the Brevicoceratidae or Acleistoceratidae of the Oncocerida early in the Devonian...

, contains three genera (as of 1964), all confined to the Middle Devonian: Tetragonoceras

Tetragonoceras

Tetragonoceras is an extinct prehistoric nautiloid genus from the nautilid family Tetragonoceratidae that lived during the Middle Devonian, found in Canada....

, Nassauoceras

Nassauoceras

Nassauoceras is a tetragonoceratid nautiloid from the Middle Devonian of Europe, the shell of which is evolute with a wide, deep umbilucus, slight dorsal impression, low arched venter, rounded ventral shoulders, and flanks that converge dorsally so as to produce a subtrigonal whorl section. Nodes...

, and Wellsoceras

Wellsoceras

Wellsoceras is a tetragonoceratid that starts off with an evolute shell in which whorls are in contact, but has a mature living chamber that diverges and becomes free. The whorl section is slightly wider than high and is faintly subquadrangular. Flanks are slightly convex and tend to comverge...

.