In the plane, the exterior angle of a vertex is $\pi -$ the standard ("interior") angle, which may be negative in some cases. The following is true for non-weird polygons:



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> The sum of the exterior angles at each vertex is a full turn ($2\pi$ radians).
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I am informally calling polygons with self-crossing edges or holes as "weird" -- please do let me know what the standard terminology is. I have seen an extension to 3-dimensional polytopes of this form:



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> The sum of the exterior angles of a polytope is $4\pi$ radians.
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In this case, the exterior angle of a vertex is $2\pi -$ (the sum of the face angles at that vertex). I have not seen a proof, but I think it is true for non-weird polytopes, and a modified version is probably true for polytopes with nonzero genus.


My question is: is there a general n-dimensional version of these properties?



An exterior angle theorem for n-dimensional polytopes?
What you call exterior angles are the curvatures at the vertices, and the result you are referring to is the combinatorial Gauss-Bonnet theorem. It says that in an angled 2-complex $K$ the sum of the curvatures at the vertices plus the sum of the curvatures at the polygonal faces is $2\pi \chi(K)$ (Euler characteristic). When your complex is piecewise Euclidean then the curvature at each face is zero implying your result in the case of polyhedra. For polyhedra, another way to see this is to perform barycentric subdivision and sum the angles of all faces to reduce your identity to $V+F-E=2$. A possible reference is [this](http://arxiv.org/abs/0910.5688) paper (theorem 4.8).


