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Euler's Formula

(There is another "Euler's Formula" about complex numbers,
this page is about the one used in Geometry and Graphs)

Euler's Formula

For any polyhedron that doesn't intersect itself, the

  • Number of Faces
  • plus the Number of Vertices (corner points)
  • minus the Number of Edges

always equals 2

This can be written: F + V - E = 2

Try it on the cube:

A cube has 6 Faces, 8 Vertices, and 12 Edges,


6 + 8 - 12 = 2

To see why this works, imagine taking the cube and adding an edge
(say from corner to corner of one face).

You will have an extra edge, plus an extra face:

7 + 8 - 13 = 2


Likewise if you included another vertex (say halfway along a line)
you would get an extra edge, too.

6 + 9 - 13 = 2.

"No matter what you do, you always end up with 2"
(But only for this type of Polyhedron ... read on!)


Example With Platonic Solids

Let's try with the 5 Platonic Solids (Note: Euler's Formula can be used to prove that there are only 5 Platonic Solids):

Name   Faces Vertices Edges F+V-E
Tetrahedron 4 4 6 2
Cube 6 8 12 2
Octahedron 8 6 12 2
Dodecahedron 12 20 30 2
Icosahedron 20 12 30 2

So, the result is always 2 ...

* But Not Always ...

Now that you see how this works, I am going to show you how it doesn't work ...!

What if I joined up two opposite corners of the icosahedron?

It is still an icosahedron (but no longer regular or convex).

In fact it looks a bit like a drum where someone has stitched the top and bottom together.

Now, there would be the same number of edges and faces ... but one less vertex!


F + V - E = 1

Oh No! It doesn't always add to 2!

The reason it didn't work was that this new shape is basically different.

All Platonic Solids (and many other solids) are like a Sphere ... you can reshape them so that they become a sphere (move their corner points, then curve their faces a bit).

But you can't do that with this new shape, because that joined bit in the middle won't let you!

Euler Characteristic

So, F+V-E can equal 2, or 1, and maybe other values, so the more general formula is

F + V - E = χ

Where χ is called the "Euler Characteristic".

Here are a few examples:

Shape   χ
Sphere 2
Torus 0
Mobius Strip 0

And the Euler Characteristic can also be less than zero.

This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so:

F + V - E = -2

In fact the Euler Characteristic is a basic idea in Topology (the study of the Nature of Space).

Donut and Coffee Cup

Lastly, this discussion would be incomplete without showing you that a Donut and a Coffee Cup are really the same!

Well, they can be deformed into one another.

We say the two objects are "homeomorphic" (from Greek homoios = identical and morphe = shape)

Just like the platonic solids are homeomorphic to the sphere.

(Animation courtesy of Wikipedia User:Kieff)