Angular defect

In geometry, the amount by which the sum of the angles at a vertex of a polyhedron falls short of a full circle is the defect of the vertex. If the sum of the angles exceeds a full circle, then the defect is negative.

For example, the defect of any of the vertices of a cube is a right angle. The defect of any of the vertices of a dodecahedron (in which three regular pentagons meet at each vertex) is 36 degrees, or π/5 radians, or 1/10 of a circle.

Descartes' theorem on the "total defect" of a polyhedron states the if the polyhedron is homeomorphic to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720 degrees or 4π radians).