A plane can mean an airplane that flies in the sky, or a flat, level surface in geometry.
The flat-surface sense comes from Latin 'planum' meaning 'flat surface', from 'planus' meaning 'flat'. The aircraft sense is a short form of 'aeroplane', from Greek 'aero' (air) and 'planos' (wandering).
The 'plane' in geometry and the 'plane' you fly on are not as related as they look—one is about flatness, the other about wandering through the air. But airplane wings do rely on carefully shaped surfaces, so in a way, geometry still lifts you off the ground.
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