Editing Gauss–Bonnet theorem (section)

=== 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]].