# Triangle, defect of a

The amount by which the sum of the internal angles of a triangle on the Lobachevskii plane is short of two right angles. The defect of a triangle is proportional to its area. The maximum value of the defect of a triangle equals two right angles if all the vertices of the triangle are points at infinity.

#### Comments

The corresponding notion for geodesic triangles on general surfaces of negative curvature is contained in the Gauss–Bonnet integral formula (cf. Gauss–Bonnet theorem).

For a triangle in spherical geometry one has the opposite effect that the sum of the angles is greater than two right angles, the spherical excess or angular excess.

The defect of a triangle is also called angular defect or hyperbolic defect.

#### References

[a1] | M. Berger, "Geometry" , II , Springer (1987) pp. §19.5 |

[a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. §16.5 |

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Triangle, defect of a.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Triangle,_defect_of_a&oldid=17691