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

When working exclusively in two-dimensional Euclidean space, the definite article is used, so *the* plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, or, in other words, in the plane.

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.^{[1]} He selected a small core of undefined terms (called *common notions*) and postulates (or axioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the *Elements*, it may be thought of as part of the common notions.^{[2]} Euclid never used numbers to measure length, angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane.

A plane is a ruled surface.

This section is solely concerned with planes embedded in three dimensions: specifically, in **R**^{3}.

This section is solely concerned with planes embedded in three dimensions: specifically, in **R**^{3}.

In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following:

- Three non-collinear points (points not on a single line).
- A line and a point not on that line.
- Two distinct but intersecting lines.
- Two distinct but parallel lines.

The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues:

- Two distinct planes are either parallel or they intersect in a line.
- A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
- Two distinct lines perpendicular to the same plane must be parallel to each other.
- Two distinct planes perpendicular to the same line must be parallel to each other.

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let **r**_{0} be the position vector of some point *P*_{0} = (*x*_{0}, *y*_{0}, *z*_{0}), and let * n* = (

This section is solely concerned with planes embedded in three dimensions: specifically, in **R**^{3}.

which is the *point-normal* form of the equation of a plane.^{[3]} This is just a linear equation

where

This plane can also be described by the "point and a normal vector" prescription above. A suitable normal vector is given by the cross product

and the point **r**_{0} can be taken to be any of the given points **p**_{1},**p**_{2} or **p**_{3}^{[6]} (or any other point in the plane).

For a plane and a point not necessarily lying on the plane, the shortest distance from to the plane is