A plane is a perfectly flat surface that has no thickness and extends forever in every direction, like a tabletop that never ends. It is one of the undefined terms of geometry, along with point and line. Any three points that are not on one line lie in exactly one plane.
A plane is drawn as a parallelogram, but it goes on forever. Three points not on one line fix exactly one plane, and two planes that cross meet in a line.
Formal definition and notation
In Euclidean geometry, a plane is an undefined two-dimensional object described by its properties: through any three noncollinear points there is exactly one plane; if two points lie in a plane, the whole line through them lies in that plane; and two distinct planes either never meet (parallel planes) or meet in exactly one line. In coordinates, a plane is the set of points (x, y, z) with ax + by + cz = d, where a, b and c are not all 0.
Symbol
Read it as
Meaning
plane P
plane P
a plane named by one capital letter, often written in script
plane ABC
plane A B C
a plane named by three of its points that are not on one line
z = 4
z equals 4
in 3D coordinates, the flat plane of all points 4 units above the xy-plane
Key facts
A plane has two dimensions (length and width) but no thickness. A drawing of a plane is a parallelogram, but the real plane has no edges.
Three noncollinear points determine exactly one plane. That is why a three-legged stool never wobbles.
Points or lines in the same plane are coplanar. Two lines in one plane either cross or are parallel; lines that are not coplanar and never meet are skew lines.
Two different planes are either parallel (they never meet) or they intersect in exactly one line, like two walls meeting at a corner.
Points A, B and C are not on the same line. How many planes contain all three points?
Through two points there are many planes, like pages turning around a book’s spine.
Adding a third point that is off the line through the first two pins one page in place.
So three noncollinear points lie in exactly one plane.
Answer: Exactly 1 plane contains A, B and C.
Example 2: Planes in a cube
Core
Each of the 6 faces of a cube lies in a plane. How many pairs of these planes are parallel, and how many lines do the other pairs meet in?
Opposite faces never meet, so they are parallel planes: top-bottom, front-back, left-right. That is 6 ÷ 2 = 3 pairs.
Count all pairs of faces: each of the 6 faces pairs with 5 others, and each pair is counted twice, so 6 × 52 = 15 pairs.
Every pair that is not parallel meets in a line, which is an edge of the cube: 15 − 3 = 12.
Check: a cube has 12 edges.
Answer: There are 3 pairs of parallel planes, and the other pairs meet in 12 lines (the 12 edges).
Example 3: Planes determined by four points
Test-style
Points A, B, C and D are the corners of a triangular pyramid, so no three are on one line and all four are not in one plane. How many planes do they determine?
Each plane needs 3 noncollinear points, so count the groups of 3 points.
The groups are ABC, ABD, ACD and BCD.
No two groups give the same plane, because all four points are not coplanar.
Each group is one face of the pyramid.
Answer: The four points determine 4 planes, one for each face of the pyramid.
Common mistakes
✗ Mistake“A plane is a rectangle” or “plane P ends at its edges”.
✓ InsteadA plane has no edges; the parallelogram is only a picture of part of it.
Like a line, a plane extends forever. Textbooks draw a slanted parallelogram so the page looks three-dimensional.
✗ MistakeAny three points determine one plane, including three points on one line.
✓ InsteadThree points on one line lie in infinitely many planes. Only three noncollinear points fix one plane.
A line is like a hinge: many planes can turn around it. The third point must be off the line to lock one plane in place.
✗ MistakeTwo lines that do not meet must be parallel.
✓ InsteadThey are parallel only if they are in the same plane. Non-coplanar lines that never meet are skew.
In space, a line along the floor and a line across the ceiling going the other way never meet, yet they are not parallel.
A line is one-dimensional: it has length but no width. A plane is two-dimensional: it has length and width. A plane contains infinitely many lines.
Planevs.Plane figure
A plane figure is a flat shape with a boundary, such as a triangle or circle, drawn in a plane. The plane itself has no boundary.
Quick check: 3 questions
Question 1 of 3
Which is the best description of a plane?
Show answer and explanation
Answer: A. A plane is an endless flat surface with no thickness. A four-sided shape is a quadrilateral, an endless straight path is a line, and a solid with flat faces is a polyhedron.
Question 2 of 3
Two different planes intersect. What is their intersection?
Show answer and explanation
Answer: B. Two distinct planes that are not parallel meet in exactly one line, like a wall meeting the floor. A point is where two lines cross, and planes have no endpoints, so the intersection cannot be a ray.
Question 3 of 3
How many planes contain three points that lie on one line?
Show answer and explanation
Answer: D. Points on one line act like a hinge, and infinitely many planes turn around it. Exactly 1 plane is the answer only when the three points are not on one line.
Plane: frequently asked questions
Why is a plane an undefined term?
Every definition uses other words, so geometry has to start somewhere. Point, line and plane are left undefined and described by how they behave instead: a plane is flat, has no thickness, and three noncollinear points fix exactly one. Every other figure, such as an angle, triangle or circle, is then defined using these three terms.
How do you name a plane?
Use one capital letter, usually in script, such as plane P, or three points in the plane that are not on one line, such as plane ABC. The three points can be listed in any order, so plane ABC, plane BCA and plane CAB all name the same plane.
What is the difference between a plane and a coordinate plane?
A coordinate plane is a plane with two number lines (the x-axis and y-axis) drawn on it at right angles. The axes let you name every point with an ordered pair. The plane itself is just the endless flat surface; the grid and axes are added tools for locating points.
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