Key takeaway: A point, line and plane are the three undefined terms of geometry: a point names one exact location, a line runs straight without end both ways, and a plane is a flat sheet without edges.
Grade 10 – GeometryG-CO.A.1Geometry and Measurement
Definition of Point, Line and Plane
The point, line and plane trio is where all of geometry begins. Nobody can define them using simpler words, so they are called undefined terms and are explained through pictures instead.
Picture the tip of a sharpened pencil touching paper: that is a point, a location with no width. Stretch a taut string in both directions without stopping: that is a line. Imagine a sheet of glass with no thickness and no edges: that is a plane. Segments, angles, triangles and cubes are all assembled from these three ideas.
Notation
A point is named with one capital letter, such as \(A\). A line is named by two of its points, \(\overleftrightarrow{AB}\), or by one lowercase letter. A plane is named by a capital script letter, such as plane \(\mathcal{P}\), or by three noncollinear points, such as plane \(ABC\).
Key facts
- Two points determine exactly one line; three noncollinear points determine exactly one plane.
- Points on the same line are collinear; points in the same plane are coplanar.
- Two different lines meet in at most one point; two different planes meet in a line.
- A drawing of a plane shows only a small piece, since the real plane has no edges.
Point, Line and Plane example
Four points \(T\), \(U\), \(V\) and \(W\) are marked on a flat sheet of paper, and no three of them lie on one line. How many different lines pass through at least two of the points?
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Two points determine exactly one line, so count the pairs of points.
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List them: \(TU\), \(TV\), \(TW\), \(UV\), \(UW\), \(VW\).
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No three points are collinear, so no two pairs share the same line.
There are 6 different lines, and all four points lie in the single plane of the paper.
Common mistake
Watch out
When reviewing point, line and plane facts, Omar says any three points determine exactly one plane, so points \(E\), \(F\) and \(G\) on the same straight line must give one plane.
Why it is wrong
The rule needs three noncollinear points. If \(E\), \(F\) and \(G\) are collinear, infinitely many planes contain them, like the pages of an open book all sharing the spine. A third point off the line is needed to fix one plane.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement