In Euclidean geometry, a **plane** is a flat two-dimensional surface that extends indefinitely.
Planes often arise as subspaces of three-dimensional space, as with one of a room's walls, infinitely extended.

Main article: Euclidean geometry |

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. The Euclidean plane equipped with a chosen Cartesian coordinate system is called a *Cartesian plane*; a non-Cartesian Euclidean plane equipped with a polar coordinate system would be called a *polar plane*.

A plane is a ruled surface.

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* = (

The dot here means a dot (scalar) product.

Expanded this becomes

which is the

where

which is the expanded form of

In mathematics it is a common convention to express the normal as a unit vector, but the above argument holds for a normal vector of any non-zero length.

Conversely, it is easily shown that if *a*, *b*, *c*, and *d* are constants and *a*, *b*, and *c* are not all zero, then the graph of the equation

is a plane having the vector

Thus for example a regression equation of the form *y* = *d* + *ax* + *cz* (with *b* = −1) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Alternatively, a plane may be described parametrically as the set of all points of the form

where s and t range over all real numbers, **v** and **w** are given linearly independent vectors defining the plane, and **r**_{0} is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors **v** and **w** can be visualized as vectors starting at **r**_{0} and pointing in different directions along the plane. The vectors **v** and **w** can be perpendicular, but cannot be parallel.

Let **p**_{1} = (*x*_{1}, *y*_{1}, *z*_{1}), **p**_{2} = (*x*_{2}, *y*_{2}, *z*_{2}), and **p**_{3} = (*x*_{3}, *y*_{3}, *z*_{3}) be non-collinear points.

The plane passing through **p**_{1}, **p**_{2}, and **p**_{3} can be described as the set of all points (*x*,*y*,*z*) that satisfy the following determinant equations:

To describe the plane by an equation of the form , solve the following system of equations:

This system can be solved using Cramer's rule and basic matrix manipulations. Let

If *D* is non-zero (so for planes not through the origin) the values for *a*, *b* and *c* can be calculated as follows:

These equations are parametric in *d*. Setting *d* equal to any non-zero number and substituting it into these equations will yield one solution set.

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

In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane.

It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on the shifted plane that is closest to the origin. The resulting point has Cartesian coordinates :

- .

In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.

Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in computer graphics, motion planning, and collision detection.When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:

Let *S* be a sphere with center *O*, *P* a plane which intersects *S*. Draw *OE* perpendicular to *P* and meeting *P* at *E*. Let *A* and *B* be any two different points in the intersection. Then *AOE* and *BOE* are right triangles with a common side, *OE*, and hypotenuses *AO* and *BO* equal. Therefore, the remaining sides *AE* and *BE* are equal. This proves that all points in the intersection are the same distance from the point *E* in the plane *P*, in other words all points in the intersection lie on a circle *C* with center *E*.^{[7]} This proves that the intersection of *P* and *S* is contained in *C*. Note that *OE* is the axis of the circle.

Now consider a point *D* of the circle *C*. Since *C* lies in *P*, so does *D*. On the other hand, the triangles *AOE* and *DOE* are right triangles with a common side, *OE*, and legs *EA* and *ED* equal. Therefore, the hypotenuses *AO* and *DO* are equal, and equal to the radius of *S*, so that *D* lies in *S*. This proves that *C* is contained in the intersection of *P* and *S*.

As a corollary, on a sphere there is exactly one circle that can be drawn through three given points.^{[8]}

The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.^{[9]}