A vertex in maths is a point where two or more lines, line segments, edges, or curves meet. The plural of vertex is vertices. The corner of a square - where two sides come together - is a vertex.
Vertices appear across geometry and algebra: in angles, in 2D polygons, in 3D solids, in parabolas, and in the network diagrams of graph theory. Each context uses the same idea - a single point where two or more elements meet.
Formal Definition of a Vertex
A vertex is the endpoint of two or more line segments, rays, or edges that meet at a single point. The two sides must end at the vertex - they do not pass through it. If two lines cross and continue past each other, that point is an intersection, not a vertex.
In angle notation, the vertex is named by the middle letter. In β ABC, point B is the vertex. The two rays start at B and extend outward through points A and C.
Vertex in 2D Shapes (Polygons)
In a polygon, a vertex is the point where two sides meet. The number of vertices in any polygon equals its number of sides. A triangle has 3 vertices. A pentagon has 5. Curved shapes like circles and ovals have no vertices because they have no straight sides meeting at a point.
2D Shape | Sides | Vertices |
|---|---|---|
Triangle | 3 | 3 |
Quadrilateral (square, rectangle) | 4 | 4 |
Pentagon | 5 | 5 |
Hexagon | 6 | 6 |
Heptagon | 7 | 7 |
Octagon | 8 | 8 |
Decagon | 10 | 10 |
Circle | 0 | 0 |
Vertex in 3D Shapes (Polyhedra)
In a solid shape, a vertex is the point where three or more edges meet. A cube has 8 vertices, with three edges meeting at each one.
The number of vertices in any convex polyhedron can be found using Euler's formula:
V β E + F = 2
where V is the number of vertices, E is the number of edges, and F is the number of faces. A cube has 12 edges and 6 faces, so V = 2 + E β F = 2 + 12 β 6 = 8.
3D Shape | Faces | Edges | Vertices |
|---|---|---|---|
Cube | 6 | 12 | 8 |
Cuboid (rectangular prism) | 6 | 12 | 8 |
Tetrahedron | 4 | 6 | 4 |
Square pyramid | 5 | 8 | 5 |
Triangular prism | 5 | 9 | 6 |
Cone | 2 | 1 | 1 (apex) |
Cylinder | 3 | 2 | 0 |
Sphere | 1 | 0 | 0 |
Apex vs Vertex
The apex of a shape is its topmost vertex β the single point above the base in a pyramid or cone. Every apex is a vertex, but not every vertex is an apex. A cube has 8 vertices and no apex, because no single vertex sits above all the others. A square pyramid has 5 vertices, and one of them β the tip above the square base β is the apex.
Vertex of an Angle
The vertex of an angle is the point where its two rays begin. Both rays share that common endpoint and extend outward from it.
When an angle is named with three letters, the vertex letter sits in the middle. For β PQR, the vertex is at Q. When the context is clear, an angle can also be named by its vertex letter alone β β Q refers to the same angle.
Vertex of a Parabola
In the graph of a quadratic equation, the vertex is the highest or lowest point of the parabola β the point where the curve changes direction. A parabola opening upward has a vertex at its minimum. One opening downward has a vertex at its maximum.
For a parabola written in vertex form:
y = a(x β h)Β² + k
the vertex sits at the point (h, k).
For the standard form y = axΒ² + bx + c, the x-coordinate of the vertex is given by x = βb/2a. Substituting that x-value back into the equation gives the y-coordinate.
The parabola vertex is called a vertex for the same reason a polygon vertex is β both are points where direction turns. (The etymology section below explains the connection.)
Vertex in Graph Theory
In graph theory, a vertex is a node in a network β a point connected to other points by edges. A road map is a familiar example: each city is a vertex, each road between two cities is an edge. Computer networks, family trees, and circuit diagrams all use vertices and edges to model connections.
In graph theory, vertices are sometimes called nodes. The two words mean the same thing.
Where the Word "Vertex" Comes From
Vertex is from the Latin word vertere, meaning "to turn." A polygon vertex is where the boundary turns from one side to the next. A parabola vertex is where the curve turns from rising to falling, or from falling to rising. Both meanings come from the same root β and that's why mathematicians use the word in both contexts.
Vertex vs Other Geometry Terms
Several words describe related but distinct ideas. The table below maps the differences.
Term | Meaning | How It Relates to Vertex |
|---|---|---|
Edge | A line segment where two faces meet (in 3D) or one side of a polygon (in 2D) | Edges connect vertices; vertices are the endpoints of edges |
Face | A flat surface of a solid shape | Faces meet at edges; edges meet at vertices |
Apex | The topmost vertex of a shape with a base | A specific kind of vertex (for example, the tip of a cone) |
Corner | Informal name for a vertex | Same idea, casual word |
Node | Vertex in graph theory | Same concept under a different name |
Intersection | The point where two lines cross | Not a vertex β at a vertex, lines end; at an intersection, they continue past |
Common Confusions
Vertex vs. intersection. At a vertex, the two sides end. At an intersection, the two lines continue past each other. The point where two crossing roads meet on a map is an intersection, not a vertex.
Curved shapes and vertices. Circles, spheres, and cylinders have no vertices. They have no straight edges meeting at a point. A cone has one vertex β the apex β even though most of the shape is curved.
Edges and vertices in 3D shapes. An edge is a line; a vertex is a point. A cube has 12 edges and 8 vertices β three edges meet at each vertex.
Was this article helpful?
Your feedback helps us write better content
