Euler's Theorom

Question

Proof of Eular's Theorom F+V-E=2

Answer 1

Euler's Formula states that there is a unique relationship between the number of faces, vertices, and edges of every polyhedron (F + V - E = 2, where "F" is the total number of faces, "V" is the total number of vertices, and "E" is the total number of edges).  A polyhedron is a solid, 3-dimensional, closed figure.  Let's review the five regular, convex Platonic solids and their characteristics....

 

1.  Tetrahedron:  4 faces, 4 vertices, 6 Edges

     F + V - E = 2.......Euler's Formula

          4+4-6=2........substitute for "F, V, E"

              8-6=2........simplify

                 2=2........true, check, formula proven

 

2. Cube: 6 faces, 8 vertices, 12 Edges
    F + V - E = 2.......Euler's Formula
       6+8-12=2........substitute for "F, V, E"

         14-12=2........simplify
                2=2........true, check, formula proven

 

Now continue this exercise for the remaining Platonic solids listed below.  Find the number of sides, vertices and edges for each solid and substitute into Euler's Formula to verify.......

 

3.  Octahedron:

4.  Dodecahedron:

5.  Icosahedron:

 

 

For more information on Euler's Formula, perhaps you may want to review the following at...

https://www.math.ubc.ca/~cass/courses/m308-03b/projects-03b/wagner/Webpage.htm

Answer 2

Twenty of them are presented at:

https://www.ics.uci.edu/~eppstein/junkyard/euler/