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plane has 11 distinct synonyms spread across 7 of its 9 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
an aircraft that has a fixed wing and is powered by propellers or jets
“the flight was delayed due to trouble with the airplane”
(mathematics) an unbounded two-dimensional shape
“we will refer to the plane of the graph as the X-Y plane”
a level of existence or development
“he lived on a worldly plane”
a carpenter's hand tool with an adjustable blade for smoothing or shaping wood
“the cabinetmaker used a plane for the finish work”
Synonyms, antonyms and the broader and narrower relations for plane come from Princeton University’s WordNet. Full sources and licences.
make even or smooth, with or as with a carpenter's plane
“plane the top of the door”