Editing Internal and external angles (section)

==Extension to polyhedra==
{{further|Descartes' theorem on total angular defect}}

Consider a [[polyhedron]] that is [[Homeomorphism|topologically equivalent]] to a [[sphere]], such as any [[convex polytope|convex polyhedron]].  Any vertex of the polyhedron will have several [[facet (geometry)|facets]] that meet at that vertex.  Each of these facets will have an interior angle at that vertex and the sum of the interior angles at a vertex can be said to be the interior angle associated with that vertex of the polyhedron.  The value of {{math|2''π''}} radians (or 360 degrees) minus that interior angle can be said to be the exterior angle associated with that vertex, also known by other names such as [[angular defect]].  The sum of these exterior angles across all vertices of the polyhedron will necessarily be {{math|4''π''}} radians (or 720 degrees), and the sum of the interior angles will necessarily be {{math|2''π''(''n'' − 2)}} radians (or {{math|360(''n'' − 2)}} degrees) where {{mvar|n}} is the number of vertices.  A proof of this can be obtained by using the formulas for the sum of interior angles of each facet together with the fact that the [[Euler characteristic]] of a sphere is 2.

For example, a [[rectangular solid]] will have three rectangular facets meeting at any vertex, with each of these facets having a 90° internal angle at that vertex, so each vertex of the rectangular solid is associated with an interior angle of {{math|1=3 × 90° = 270°}} and an exterior angle of {{math|1=360° − 270° = 90°}}.  The sum of these exterior angles over all eight vertices is {{math|1=8 × 90° = 720°}}.  The sum of these interior angles over all eight vertices is {{math|1=8 × 270° = 2160°}}.