In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point (zero-dimensions), a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

 When working in two-dimensional Euclidean space, the definite article is used, the plane, to refer to the whole space. Many fundamental tasks in geometry, trigonometry, and graphing are performed in two-dimensional space, or in other words, in the plane. A lot of mathematics can be and has been performed in the plane, notably in the areas of geometry, trigonometry, graph theory and graphing.

In mathematics, plane geometry may refer to:

·         Euclidean geometry, the geometry of plane figures,

·         geometry of a plane

·         geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;

·         geometry of the hyperbolic plane or two-dimensional spherical geometry.