what does this formula prove?

Question

The formula 'V - E + F = 2' where v number of vertices, e number of edges and f the number of faces. What exactly is this formula trying to show us.

The formula 'V - E + F = 2' where v number of vertices, e number of edges and f the number of faces. What exactly is this formula trying to show us. Why does it equal 2, what does that mean?

Answer 1

Euler's Formula

This classic formula holds for any convex polyhedron.

The number of vertices and faces together is exactly two more than the number of edges.
Symbolically V-E+F=2, where V is the number of vertices, E is the number of edges, and F is the number of faces of the polyhedron.

See examples:-

Answer 2

It's not the 2 that's important. The formula would be just as interesting and useful if it somehow happened that V - E + F = 6, or V - E + F = -1.

It's the arrangement of plus and minus that shows us something really important. Any change in the number of edges has to be balanced by the same change in the number of (vertices plus faces). It's not easy to visualise, but it's important.

Answer 3

The formula shows that there is a connection between the number of vertices,faces and edges.
Example
Consider a cuboid
6 faces and 12 edges
V - 12 + 6 = 2
V - 6 = 2
V = 8 (is correct no. of vertices (corners))

Answer 4

Take a familiar shape such as a cube for example:

It has 6 faces, 12 edges and 8 vertices (corners). The number of faces + the number of vertices = number of edges + 2

F + V = E + 2

This is true for all convex (vertices stick out) polyhedron (many sided shapes) and is called Euler's relation after a famous mathematician

Answer 5

I guess its useful because you can rearrange it, so if you only have two of the variables you can work out the final variable. So if you knew what V & F were you could rearrange the formula into

V + F - 2 = E

Answer 6

it shows that i am hopeless at maths...