Euler's Formula is one of the cornerstones of mathematical analysis that ties together complex numbers with trigonometric functions nicely. Their profound implications, particularly those that have been delivered through the example of Euler's identity have convinced many about the unity of concepts and constants that seemed different. It is a universal tool since it appears in various places, from engineering, physics, to signal processing, thus constituting a very promising basis for understanding oscillatory behavior and complex dynamics.

Euler’s Formula Equation

Euler's formula or Euler's Identity states that for any real number x, in complex analysis is given by:

   

              

 

Where,

• x = real number

• e = base of natural logarithm

• sin x & cos x = trigonometric functions

• i = imaginary unit

Note: The expression cos x + i sin x is often referred to as cis x.

Example Question Applying Euler's Formula Equation

Question: Find the value of e i π/2.

Solution:

Euler's Formula for Polyhedrons

Euler's polyhedron formula presents the number of vertices and faces to be two more than the number of edges. We can write Euler's formula for a polyhedron as:

Faces + Vertices = Edges + 2

F + V = E + 2

Or

F + V – E = 2

 Here,

• F = number of faces

• V = number of vertices

• E = number of edges

Let us check this formula for some solids.

• Solid name        Faces (F)       Vertices (V)       Edges (E)      Result (F + V – E)

1. Cube                             6                  8                   12                 6 + 8 – 12 = 2

2. Square pyramid            5                  5                    8                  5 + 5 – 8 = 2

3. Triangular prism            5                  6                    9                  5 + 6 – 9 = 2

Example of Euler's Formula for Solids

Example: If a polyhedron is such that it has 12 faces and 30 edges, then find out its name.

Step Solution

Given,

Number of faces = F = 12

Number of edges = 30

By making use of Euler's formula for solids,

                     F + V = E + 2   

                    12 + V = 30 + 2   

                            V = 32 – 12   

                            V = 20

As from the values of F, V, and E found above, we can reason that the polyhedron could be a Dodecahedron.

Other Related Sections

NCERT Solutions | Sample Papers | CBSE SYLLABUS| Calculators | Converters | Stories For Kids | Poems for Kids| Learning Concepts | Practice Worksheets | Formulas | Blogs | Parent Resource