Math Is Fun Forum

wintersolstice

Real Member

Registered: 2009-06-06

Posts: 128

proof of sum of angle defects

Statement: The sum of the angle defects of a Polyhedron (that doesn't intersect itself) is equal to 360 (degrees) multiplied by the Euler characteristic

Proof:

the Euler characteristic is

F (Faces)
E (Edges)
V (Vertices)

lets say that F, E and V are all we know

the angle defects can be represented as

The sum of the angles around each Vertex are

now

and so on....

Add them up

now we know V so we need

This represents the sum of all the angles in all the faces

The angles in an N sided polygon are:

or

If the number of sides in each face are given by

then the the sum of the angles are:

and so on...

Adding

is the sum of the number of edges in all the faces, and since there are two faces to an edge

so substituting that in

and therefore

which simplifies to

rearranging

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Why did the chicken cross the Mobius Band?
To get to the other ...um...!!!