Identifying Vertices in 2D and 3D Shapes

A vertex is a point where two or more line segments, edges, or rays meet to form a corner. The plural of vertex is vertices. You can find vertices on flat, two-dimensional shapes as well as on solid, three-dimensional shapes. If you have ever traced the outline of a shape and felt your pencil "turn a corner," you were tracing a vertex.

In everyday shapes, vertices are the sharp points, corners, or tips. A perfectly round shape, like a circle or a sphere, has no sharp corners, so it has no vertices at all.

In a polygon, a vertex is the point where two sides meet. To identify all the vertices of a polygon, count the number of corners — this also tells you the number of sides, since each side connects two consecutive vertices.

A triangle has 3 vertices, a square or rectangle has 4, a pentagon has 5, and so on. For any simple polygon, the number of vertices always equals the number of sides. Vertices are also the points you plot on a coordinate grid when you graph a polygon, and they are the points that move whenever a shape undergoes a transformation, which is covered in more detail in the lesson on the introduction to transformations.

Three-dimensional solids are usually described using three related counts:

• Vertices — the corner points of the solid.
• Edges — the line segments where two faces meet.
• Faces — the flat (or curved) surfaces that make up the solid.

To identify the vertices of a 3D shape, look for every point where edges come together, not just the points you can see from the front. In a drawing of a cube, some vertices are hidden behind the solid, but they still count.

The table below summarizes the vertex, edge, and face counts for shapes that come up often in geometry problems.

Notice that curved solids like the cylinder and the sphere have no sharp corners, so they have zero vertices. A cylinder still has two circular edges where its curved surface meets its two flat circular faces, but nowhere on it do straight edges meet at a point.

For any polyhedron (a solid with only flat polygon faces), the number of vertices, edges, and faces always follows a simple relationship known as Euler's formula:

\( V - E + F = 2 \)

Here \(V\) is the number of vertices, \(E\) is the number of edges, and \(F\) is the number of faces. You can use this formula to check a count or to find a missing value.

Example: Checking a Cube

A cube has \(V = 8\) vertices, \(E = 12\) edges, and \(F = 6\) faces. Substituting into Euler's formula:

\( 8 - 12 + 6 = 2 \)

The formula holds true, confirming the count is consistent.

Example: Finding a Missing Count

Suppose a solid has 6 faces and 12 edges, but you are not told how many vertices it has. Using Euler's formula:

\( V - 12 + 6 = 2 \)

\( V = 2 + 12 - 6 = 8 \)

The solid has 8 vertices, which matches a cube or a rectangular prism.

Vertices are more than just corner points to count. They are the reference points you use to measure the size of a shape, such as the distance between two vertices, which is explored further in horizontal and vertical distances. Vertices are also exactly what moves when you apply a reflection across a line of symmetry, a topic covered in line symmetry. Whether you are naming a polygon, building a 3D model, or transforming a figure on a coordinate grid, correctly identifying every vertex is the first step.

• Forgetting hidden vertices on the far side of a 3D drawing.
• Confusing an edge (a line) with a vertex (a point).
• Assuming a curved solid has vertices where the curve looks sharp in a drawing, even though it has none.
• Miscounting vertices on a pyramid by forgetting to include the apex, the single point at the top.

Related lessons

• Geometry Help