Directed evolution of life through millions of years, such as increasing adult body size, is one of the most intriguing patterns displayed by fossil lineages. Processes and causes of such evolutionary trends are still poorly understood. Ammonoids (externally shelled marine cephalopods) are well known to have experienced repetitive morphological evolutionary trends of their adult size, shell geometry and ornamentation. This study analyses the evolutionary trends of the family Acrochordiceratidae Arthaber, 1911 from the Early to Middle Triassic (251–228 Ma). Exceptionally large and bed-rock-controlled collections of this ammonoid family were obtained from strata of Anisian age (Middle Triassic) in north-west Nevada and north-east British Columbia. They enable quantitative and statistical analyses of its morphological evolutionary trends. This study demonstrates that the monophyletic clade Acrochordiceratidae underwent the classical evolute to involute evolutionary trend (i.e. increasing coiling of the shell), an increase in its shell adult size (conch diameter) and an increase in the indentation of its shell suture shape. These evolutionary trends are statistically robust and seem more or less gradual. Furthermore, they are nonrandom with the sustained shift in the mean, the minimum and the maximum of studied shell characters. These results can be classically interpreted as being constrained by the persistence and common selection pressure on this mostly anagenetic lineage characterized by relatively moderate evolutionary rates. Increasing involution of ammonites is traditionally interpreted by increasing adaptation mostly in terms of improved hydrodynamics. However, this trend in ammonoid geometry can also be explained as a case of Cope’s rule (increasing adult body size) instead of functional explanation of coiling, because both shell diameter and shell involution are two possible paths for ammonoids to accommodate size increase.