Etymology
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plane (n.1)

"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.

plane (n.2)

1908, short for aeroplane (see airplane).

plane (n.3)

"tool for smoothing surfaces," mid-14c., from Old French plane, earlier plaine (14c.) and directly from Late Latin plana, back-formation from planare "make level," from Latin planus "level, flat, smooth" (from PIE root *pele- (2) "flat; to spread").

plane (n.4)

"tree of the genus Platanus," native to Persia and the Levant, late 14c., from Old French plane, earlier plasne (14c.), from Latin platanus, from Greek platanos, earlier platanistos "plane tree," a species from Asia Minor, associated with platys "broad" (from PIE root *plat- "to spread") in reference to its leaves. Applied since 1778 in Scotland and northern England to the "sycamore" maple (mock-plane), whose leaves somewhat resemble those of the true plane tree. Compare sycamore.

plane (v.1)

"to make smooth," early 14c., originally in a figurative sense, "to gloss over, explain away;" mid-14c. as "to make smooth or even" (especially by use of a plane (n.3)), from Old French planer "to smooth, level off; wipe away, erase" (12c.) and directly 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.

plane (v.2)

"soar, glide on motionless wings," early 15c., planen, from Old French planer "to hover (as a bird), to lie flat," from plan (n.) "plane," or perhaps via Medieval Latin; in either case from Latin planum "flat surface" (from PIE 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 attested by 1913. Related: Planed; planing.

plane (adj.)

"having the characteristics of a plane," 1560s, from French plan, from Latin planus "flat, level, even" (see plane (n.1)).

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Definitions of plane
1
plane (n.)
an aircraft that has a fixed wing and is powered by propellers or jets;
Synonyms: airplane / aeroplane
plane (n.)
(mathematics) an unbounded two-dimensional shape;
any line joining two points on a plane lies wholly on that plane
we will refer to the plane of the graph as the X-Y plane
Synonyms: sheet
plane (n.)
a level of existence or development;
he lived on a worldly plane
plane (n.)
a power tool for smoothing or shaping wood;
Synonyms: planer / planing machine
plane (n.)
a carpenter's hand tool with an adjustable blade for smoothing or shaping wood;
the cabinetmaker used a plane for the finish work
Synonyms: carpenter's plane / woodworking plane
2
plane (v.)
cut or remove with or as if with a plane;
Synonyms: shave
plane (v.)
travel on the surface of water;
Synonyms: skim
plane (v.)
make even or smooth, with or as with a carpenter's plane;
plane the top of the door
3
plane (adj.)
having a surface without slope, tilt in which no part is higher or lower than another;
a plane surface
Synonyms: flat / level
From wordnet.princeton.edu