"flat surface," 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." Figurative sense is attested from 1850. As an adjective from 1660s.