Vertices — Definition, Formula & Examples
Vertices are the corner points where two or more edges meet on a shape or solid. A single corner point is called a vertex, and vertices is the plural form.
A vertex of a polygon is a point at which two sides intersect. For a polyhedron, a vertex is a point at which three or more edges converge. The set of all such points constitutes the vertices of the figure.
Key Formula
Where:
- = Number of vertices
- = Number of edges
- = Number of faces
How It Works
To count the vertices of any shape, look for the sharp corners where straight edges come together. On a flat (2D) shape like a triangle, each corner where two sides meet is a vertex. On a 3D solid like a cube, each corner where three edges meet is a vertex. Knowing the number of vertices helps you classify shapes and use important formulas like Euler's formula for polyhedra.
Worked Example
Problem: Count the vertices of a cube and verify Euler's formula.
Step 1: Count the corners of a cube. A cube has 8 corners, so it has 8 vertices.
Step 2: Count the edges. A cube has 12 edges.
Step 3: Count the faces. A cube has 6 flat square faces.
Step 4: Plug into Euler's formula to check.
Answer: A cube has 8 vertices, and Euler's formula confirms: 8 − 12 + 6 = 2.
Another Example
Problem: How many vertices does a triangular pyramid (tetrahedron) have?
Step 1: Picture a triangular pyramid. It has a triangular base and three triangular sides that meet at a point on top.
Step 2: Count the corners on the base. The triangular base has 3 vertices.
Step 3: Count the apex. There is 1 additional vertex at the top.
Step 4: Add them together.
Answer: A tetrahedron has 4 vertices.
Visualization
Why It Matters
Counting vertices is one of the first skills you practice in elementary geometry and it stays useful through high school and beyond. In courses on 3D geometry and graph theory, vertices help define the structure of shapes, networks, and computer models. Architects, game designers, and engineers all rely on vertex data when building 3D models.
Common Mistakes
Mistake: Confusing vertices with edges
Correction: Vertices are the corner points; edges are the line segments that connect two vertices. Think: vertices are dots, edges are lines.
Mistake: Writing "vertexes" instead of "vertices"
Correction: The correct plural of vertex is vertices (from Latin). While "vertexes" occasionally appears in informal writing, "vertices" is the standard mathematical term.