Mathematics A surface containing all the straight lines that connect any two points on it. A flat or level surface. A level of development, existence, or achievement: scholarship on a high plane. An airplane or hydroplane. A supporting surface of an airplane; an airfoil or wing. Mathematics Of or being a figure lying in a plane: a plane curve. Flat; level. See Synonyms at level. A carpenter's tool with an adjustable blade for smoothing and leveling wood. A trowel-shaped tool for smoothing the surface of clay, sand, or plaster in a mold. To smooth or finish with or as if with a plane. To remove with a plane: plane off the rough edges on a board. To work with a plane. To act as a plane. To rise partly out of the water, as a hydroplane does at high speeds. To soar or glide. To travel by airplane. The plane tree.