A plane is a two dimensional space with infinite length and width, and no thickness. In Euclidean geometry, a plane is considered to be "flat". In other geometries, a plane may or may not be flat.
In Euclidean geometry, a plane can be defined by two distinct lines, by three non-collinear points, or by one line and a point not on the line.
Plane geometry is the geometry of objects in 2 dimensional space.
Two or more geometric objects are coplanar if there is a plane that contains all of them. Geometric objects are non-coplanar if there does not exist a plane that contains all of them.
A geometric object is planar if it can be entirely contained within one plane. A geometric object is non-planar if it can not be entirely contained within one plane.
Two planes in a 3-space either intersect or are parallel. If two planes are parallel, a line that is perpendicular to the one of the planes is also perpendicular to the other. If two planes intersect, the intersection forms a line. The angle between two intersecting planes is called a dihedral angle. If the dihedral angle between two planes is a right angle, then the planes are perpendicular.
Planes are used in many contexts. Here are some of them.
Name | Illustration | Description | ||
---|---|---|---|---|
Cartesian Plane | A plane divided into four quadrants by two perpendicular number lines. The vertical number line represents a dependent variable and the horizontal number line represents the independent variable. | |||
Rectangular coordinate plane | ||||
Complex Plane | A plane divided into four quadrants by two perpendicular number lines. Complex numbers can be plotted on a complex plane. The vertical number line represents the imaginary part of the complex number, and the horizontal number line represents the real part. | |||
Argand diagram | ||||
Half-plane | A half-plane is a part of a plane cut off by a line. | |||
Inclined plane |
| A plane that intersects a reference plane. The reference plane is usually horizontal. | ||
Metric plane | Any two dimensional space which has a unit of measure for each dimension and a way to find the distance between any two points in the plane. Cartesian coordinates and Polar coordinates are examples of metric planes. | |||
Table 1: Types of planes. |
Some of the postulates and theorems dealing with planes are found in table 2. Click on the blue buttons in the manipulatives and drag them to change the figures.
Name | Manipulative | Description |
---|---|---|
Minimum Plane Postulate | A plane contains at least 3 non-collinear points. | |
Unique Plane Postulate | There is exactly one plane that passes through any 3 non-collinear points. | |
Same Plane Postulate | If two points lie in a plane, then the entire line joining those points lies in the same plane. | |
Intersecting Planes Postulate | If two planes intersect, then their intersection is a line. | |
Point, Line, Plane Theorem | If a point lies outside a line, then there is exactly one plane containing both the point and the line. | |
Intersecting Lines Plane Theorem | If two lines intersect, then there is exactly one plane containing both lines. | |
Table 2: Postulates and Theorems of Planes. |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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