WordNet sense 1 ((mathematics) an unbounded two-dimensional shape):

sheet

WordNet sense 2 (a power tool for smoothing or shaping wood):

planer, planing machine

WordNet sense 3 (a carpenter's hand tool with an adjustable blade for smoothing or shaping wood):

carpenter's plane, woodworking plane

WordNet sense 4 (an aircraft that has a fixed wing and is powered by propellers or jets):

airplane, aeroplane

WordNet sense 5 (travel on the surface of water):

skim

WordNet sense 6 (cut or remove with or as if with a plane):

shave

WordNet sense 7 (having a surface without slope, tilt in which no part is higher or lower than another):

flat, level

WordNet sense 8 (being level or straight or regular and without variation as e.g. in shape or texture; or being in the same plane or at the same height as something else (i.e. even with)):

even