"flat surface, simplest of all geometrical surfaces," c. 1600, from Latin planum "flat surface, plane, level, plain," noun use of neuter of adjective planus "flat, level, even, plain, clear," from PIE *pla-no- (source also of Lithuanian plonas "thin;" Celtic *lanon "plain;" perhaps also Greek pelanos "sacrificial cake, a mixture offered to the gods, offering (of meal, honey, and oil) poured or spread"), suffixed form of root *pele- (2) "flat; to spread."
Introduced (perhaps by influence of French plan in this sense) to differentiate the geometrical senses from plain, which in mid-16c. English also meant "geometric plane." The figurative sense, in reference to inanimate things, is attested from 1850.
1610s, American Spanish, "prairie; treeless, level plain," especially that of South America north of the Amazon, from noun use of Spanish llano "plain, even, level, smooth," ultimately from Latin planus "smooth, flat, level" (from PIE root *pele- (2) "flat; to spread"). Hence llanero "Latin-American cowboy" (1819), literally "plainsman."
late 14c., pusillanimite, "timidity, faint-heartedness, lack of the spirit of courage or fortitude," from Old French pusillanimité (14c.) and directly from Church Latin pusillanimitatem (nominative pusillanimitas) "faint-heartedness," from Latin pusillanimis "faint-hearted, having little courage" (see pusillanimous).