Generally, the term plane, together with point, line, and solid, is considered an undefined term. Every definition in mathematics attempts to use simpler and better understood terms to define more complex ones. As the terms to be defined become ever simpler, this eventually becomes impossible. The simplest terms are so well understood that there is little sense in attempting a formal definition, since often times the term itself must be used in the definition. Notice that the definition attributed to Euclid relies on an intuitive understanding of the terms point, line, straight, and surface. A plane is infinite in extent, both in length and width, so that flat physical objects are represented mathematically by some portion of a plane. A plane has only width and length. It has no thickness. While a plane is strictly two dimensional, so is the curved surface of a solid such as a sphere. In order to distinguish between curved surfaces and planes, Euclid devised a definition for plane similar to the following: given two points on a surface, the surface is planar if every point on the straight line that connects these two points is also on the surface. Plane is a term used in mathematics (especially geometry) to express, in abstract form, the physical property of flatness. A point or line can be contained in a plane, a solid cannot. Instead, the intersection of a plane with a solid is a cross section of the solid consisting of a portion of the plane.
See also Locus.