Euler's Formula for Polyhedra

Euler's Formula for polyhedra was given by the Swiss mathematician Leonhard Euler. Euler's Formula is a condition that describes the relation between the faces, vertices, and edges of any polyhedron. Polyhedron is a plural of polyhedra which is a three-dimensional shape with flat faces. It states that the face, vertex and edges of a solid shape are related to each other through a condition called Euler's Formula: F + V − E = 2, where 'F' represents the number of faces, 'V' represents the number of vertices and 'E' represents the number of edges of a solid shape.

Table of Contents

• Euler's Formula for 5 Polyhedron
• Solved Examples on Euler's Formula

Euler's Formula for 5 Polyhedron

Let's verify the Euler's formula for following polyhedron

Solved Examples on Euler's Formula

Example 1: Verify Euler's formula for a dice.
Solution: Since dice is a cube with six face, eight vertices and twelve edges, we can apply the Euler's formula F + V − E
Therefore, the value of F, V and E are:
F = 6
V = 8
E = 12

F + V − E = 2 → 6 + 8 − 12 =2 

Example 2: Find the number of edges of a polyhedron with 7 faces and 10 vertices.
Solution: We know that, F = 7 and V = 10
As per Euler's formula: F + V − E = 2
By substituting the value of F and V in above equation we get:
7 + 10 − E = 2
E = 17 − 2
E = 15
Therefore, the polyhedron has 15 edges.

Example 3: Can a polyhedron have 10 faces, 15 edges and 8 vertices?
Solution: Let's use Euler's formula to verify if a polyhedron can have 10 faces, 15 edges and 8 vertices.
Given, F = 10, E = 15 and V = 8 
As per Euler's formula: F + V − E = 2 
10 + 8 − 15 ≠ 2
Here, F + V − E ≠ 2
No, the polyhedron cannot have 10 faces, 15 edges and 8 vertices.