"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.
"tool for smoothing surfaces," mid-14c., from Old French plane, earlier plaine (14c.), from Late Latin plana, back-formation from planare "make level," from Latin planus "level, flat, smooth" (from PIE root *pele-(2) "flat; to spread").
"to make smooth," early 14c., "to gloss over, explain away;" mid-14c. as "to make smooth or even," from Old French planer "to smooth, level off; wipe away, erase" (12c.), from Late Latin planare "make level," from Latin planus "level, flat" (from PIE root *pele- (2) "flat; to spread"). In early use in English often plain. Related: Planed; planing.
"soar, glide on motionless wings," early 15c., from Old French planer "to hover (as a bird), to lie flat," from plan (n.) "plane," from Latin planum "flat surface" (root *pele- (2) "flat; to spread"), on notion of bird gliding with flattened wings. Of boats, etc., "to skim over the surface of water," it is first found 1913. Related: Planed; planing.