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The word vertex (plural: vertices) shows up in almost every branch of mathematics from the sharp corners of a triangle to the turning point of a parabola, and even in the nodes of a computer network. Whether you are a student working through geometry homework, a parent helping with algebra, or just someone curious about math, this guide explains everything you need to know clearly, completely, and with examples you can actually use.
Definition, types, vertex formula, Euler’s rule, graph theory, and 5 practice problems — everything you need in one place.
1. What Is a Vertex?
A vertex is a point where two or more lines, edges, or rays meet. It is often described as a corner or a turning point in a shape or figure.
The word vertex comes from Latin, meaning crown of the head — the topmost point. In mathematics, the same idea applies: a vertex marks a critical meeting point in a shape, equation, or network.
The plural of vertex is vertices (sometimes written vertexes, though vertices is standard in math).
Vertices appear in many different areas of mathematics:
- Geometry — corners of 2D polygons and 3D shapes
- Algebra — the maximum or minimum point of a parabola
- Graph theory — nodes (points) in a network
- Angles — the endpoint where two rays meet
Vertex in 2D Geometry (Polygons)
In any polygon, a vertex is the corner where two sides (edges) meet. The number of vertices always equals the number of sides. A triangle has 3 sides and 3 vertices; a hexagon has 6 sides and 6 vertices.
Blue/green/yellow/pink dots mark the vertices of each shape. Notice: vertices = number of sides.
| Shape | Sides | Vertices |
|---|---|---|
| Triangle | 3 | 3 |
| Quadrilateral (Square, Rectangle) | 4 | 4 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 6 |
| Octagon | 8 | 8 |
| Circle | 0 (curved) | 0 |
Convex vs Concave Vertices
Not all vertices are the same. The type of vertex depends on the interior angle formed at that corner.
| Type | Interior Angle | Description | Example |
|---|---|---|---|
| Convex vertex | Less than 180° | Points outward — shape is “normal” at that corner | All corners of a square or triangle |
| Concave (reflex) vertex | Greater than 180° | Points inward — creates a “dent” in the shape | The inner corners of a star shape |
Vertices in 3D Shapes
In solid geometry, a vertex is where three or more edges meet. Three-dimensional shapes have faces, edges, dan vertices.
| 3D Shape | Vertices (V) | Edges (E) | Faces (F) |
|---|---|---|---|
| Cube | 8 | 12 | 6 |
| Tetrahedron (triangular pyramid) | 4 | 6 | 4 |
| Rectangular prism | 8 | 12 | 6 |
| Triangular prism | 6 | 9 | 5 |
| Square pyramid | 5 | 8 | 5 |
| Sphere | 0 | 0 | 1 (curved) |
| Cylinder | 0 | 2 | 3 |
| Cone | 1 | 1 | 2 |
Euler’s Formula for Polyhedra
There is a beautiful relationship between vertices, edges, and faces in any convex polyhedron. It was discovered by the mathematician Leonhard Euler:
Vertices minus Edges plus Faces always equals 2
Cube check: 8 − 12 + 6 = 2 ✓
Tetrahedron check: 4 − 6 + 4 = 2 ✓
Vertex of an Angle
When two rays share a common endpoint, that endpoint is called the vertex of the angle. The angle is named using the vertex letter in the middle — for example, angle ∠AOB has vertex at point O.
For more on angles, see our guide on the 45-degree angle.
Vertex in Graph Theory
In graph theory (a branch of mathematics used in computer science and networking), a vertex (also called a node) is any point in a network. Edges (lines) connect the vertices to each other.
Graph theory is used to model real-world systems: cities connected by roads, people connected by friendships, computers connected by cables. Each city, person, or computer is a vertex.
Key algorithms that use graph vertices include:
- Dijkstra’s algorithm — finds the shortest path between two vertices (e.g., GPS navigation)
- Breadth-First Search (BFS) — explores vertices layer by layer (e.g., social network connections)
- Depth-First Search (DFS) — explores vertices by going as deep as possible first
Vertex of a Parabola (Quadratic Equations)
In algebra, the vertex of a parabola is the single most important point on the graph of a quadratic equation. It is either the minimum (lowest point) or the maximum (highest point) of the curve.
The Vertex Formula
For any quadratic equation in standard form y = ax² + bx + c, use this formula:
Worked Example: Find the vertex of y = x² − 4x + 3
Common Student Mistakes
| Mistake | Why It’s Wrong | Correct Thinking |
|---|---|---|
| Thinking a vertex is a line | A line has infinite points; a vertex is one specific point | Vertex = a single point (a corner) |
| Confusing vertex with edge | An edge is the line between vertices; the vertex is the corner | Edges connect vertices; vertices are the endpoints |
| Thinking a cylinder has vertices | A cylinder has circular ends with no corner points | Cylinders have 0 vertices |
| Forgetting to check the sign of b | x = −b / 2a — students forget the negative sign before b | If b = −4, then −b = +4 |
| Saying vertex = angle | The vertex is the point where the angle is formed, not the angle itself | The angle is the measurement; the vertex is the location |
Practice Problems
Test your understanding. Click an answer to see if you’re right.
1. How many vertices does an octagon have?
2. Does a sphere have any vertices?
3. What is the x-coordinate of the vertex of y = 2x² + 8x + 5?
4. A rectangular prism has 8 vertices and 6 faces. Using Euler’s formula (V − E + F = 2), how many edges does it have?
5. In graph theory, what do we call the individual points in a network (like cities on a map)?
Quick Reference Summary
In 2D Shapes
Vertices = number of sides. A triangle has 3, a square has 4, a circle has 0.
In 3D Shapes
A cube has 8 vertices. Euler’s formula: V − E + F = 2.
In Angles
The vertex of an angle is the point where two rays share an endpoint.
In Parabolas
Vertex = max or min point. Formula: x = −b/(2a).
In Graph Theory
Vertices are nodes/points in a network. Edges connect them.
Convex vs Concave
Convex: interior angle <180°. Concave: interior angle >180°.
Pertanyaan yang Sering Diajukan
Does a circle have a vertex?
No. A circle is made of a single curved line with no corners. Because there are no intersecting straight edges, a circle has zero vertices.
How many vertices does a pentagon have?
A pentagon has 5 vertices. In any polygon, the number of vertices always equals the number of sides.
What is the plural of vertex?
The plural of vertex is vertices (from Latin). The word “vertexes” is also technically correct in English but “vertices” is the standard mathematical form.
Is the vertex the same as a corner?
Yes — in everyday language, corner dan vertex mean the same thing. “Vertex” is the formal mathematical term.
What does the vertex of a parabola tell you?
The vertex of a parabola is the turning point of the curve. It tells you the maximum height (for downward parabolas) or minimum value (for upward parabolas) of the quadratic function. It is crucial for solving optimization problems in science and engineering.
