Points, Lines, and Planes

Geometry · Axiom Academy

The fundamental building blocks of geometry Everything in geometry is built from three basic objects: points , lines , and planes . These are called undefined terms — we describe them rather than define them using other terms. Think of them as the atoms of geometry: the simplest pieces from which everything else is constructed. Watch them build up. A single point — just a location — stretches into a line that runs forever in both directions; the line then sweeps out into a plane: a flat surface containing that line and other points. Each object adds one dimension to the last. Point to line to plane: zero, then one, then two dimensions. Each object is one step richer than the last. A point has no size at all; a line is built from points and runs forever in two directions; a plane is a flat sheet of points stretching out in every direction. A location in space with no size — no length, width, or height. Named with a capital letter like P or A . A straight path that extends infinitely in both directions. Has infinite length but no width. Named by two points: . A flat surface that extends infinitely in all directions. Has infinite length and width but no thickness. Named with a capital letter or three non-collinear points. These three are undefined terms — geometry's starting point. Mathematicians use a compact notation for each object so a figure can be referred to precisely. Points have no dimensions — they're just locations. Lines are one-dimensional — they have only length.

This is the written version of the interactive lesson above. See the full Geometry course.