Editing Gauss–Bonnet theorem (section)

== Special cases ==
A number of earlier results in spherical geometry and hyperbolic geometry, discovered over the preceding centuries, were subsumed as special cases of Gauss–Bonnet.

=== Triangles ===
In [[spherical trigonometry]] and [[hyperbolic geometry|hyperbolic trigonometry]], the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse) amount by which its exterior angles fail to add up to 360°.

The area of a [[spherical triangle]] is proportional to its excess, by [[Girard's theorem]] – the amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°.

The area of a [[hyperbolic triangle]], conversely is proportional to its ''defect'', as established by [[Johann Heinrich Lambert]].

=== Polyhedra ===
{{main|Descartes' theorem on total angular defect}}
[[Descartes' theorem on total angular defect]] of a [[polyhedron]] is the piecewise-linear analog:
it states that the sum of the defect at all the vertices of a polyhedron which is [[homeomorphic]] to the sphere is 4{{pi}}. More generally, if the polyhedron has [[Euler characteristic]] {{math|''χ'' {{=}} 2 − 2''g''}} (where {{mvar|g}} is the genus, the "number of holes"), then the sum of the defect is {{math|2''πχ''}}. 

This is the special case of Gauss–Bonnet in which the curvature is concentrated at discrete points (the vertices). Thinking of curvature as a [[Measure (mathematics)|measure]] rather than a function, Descartes' theorem is Gauss–Bonnet where the curvature is a [[discrete measure]], and Gauss–Bonnet for measures generalizes both Gauss–Bonnet for smooth manifolds and Descartes' theorem.