Angular defect

In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess.

Classically the defect arises in two ways:

• the defect of a vertex of a polyhedron;
• the defect of a hyperbolic triangle;

and the excess also arises in two ways:

• the excess of a toroidal polyhedron.
• the excess of a spherical triangle;

In the Euclidean plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).

In modern terms, the defect at a vertex is a discrete version of the curvature of the polyhedral surface concentrated at that point, and the Gauss–Bonnet theorem gives the total curvature as ${\displaystyle 2\pi }$ times the Euler characteristic ${\displaystyle \chi =2}$, so the sum of the defects is ${\displaystyle 4\pi }$.