How to Figure How Many Vertices a Shape Has

By Will Milner
Understanding the numbers related to shapes is crucial to geometry.

Vertices or a vertex is the technical term used in geometry for the corner points of a solid shape. A technical word is used to prevent confusion that might be used if the word “corner” was used is a description of a shape. A corner might refer to the point on the shape, but then it might also refer to the corners of the faces that make up the shape. The number of vertices can be worked out simply by counting or by using Euler's formula.



Count the vertices or “corner points," the points where the edges of the shape join up. Circle each with a pencil as you count it to avoid counting any twice. Check the entire shape to make sure all the vertices have been counted.

Rearrange Euler's formula to calculate the number of vertices in any Platonic solid, tetrahedron, cube, octahedron, dodecahedron, icosahedron. Euler's formula is usually presented as follows: Faces + Vertices - Edges = 2 However, the formula can be rearranged to make the number of vertices the subject of the formula.

Rearrange the formula as follows: Add the Edges to each side of the equation to get: Faces + Vertices = Edges + 2 Now subtract the Faces from each side of the equation to get: Vertices = Edges + 2 - Faces

Use this equation to find the vertices from the number of faces and edges as follows: Add 2 to the number of edges and subtract the number of faces. For example, a cube has 12 edges. Add 2 to get 14, minus the number of faces, 6, to get 8, which is the number of vertices.

Tip

Only use Euler's equation for the Platonic solids listed, not for other shapes. For these you will have to count.

About the Author

Will Milner started writing in 2005 for the University of Sheffield newspaper "Steel Press" and continues to write for the Sheffield-based magazine "Now Then." He gained a Bachelor of Arts in philosophy from the University of Sheffield.