Sum of angles of a triangle

A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.

The sum can be computed directly using the definition of angle based on the dot product and trigonometric identities, or more quickly by reducing to the two-dimensional case and using Euler's identity.

It was unknown for a long time whether other geometries exist, for which this sum is different.

The influence of this problem on mathematics was particularly strong during the 19th century.

Ultimately, the answer was proven to be positive: in other spaces (geometries) this sum can be greater or lesser, but it then must depend on the triangle.

Its difference from 180° is a case of angular defect and serves as an important distinction for geometric systems.

[1] In the presence of the other axioms of Euclidean geometry, the following statements are equivalent:[2] An easy formula for these properties is that in any three points in any shape, there is a triangle formed.

So, ∠A + ∠B + ∠C = 360° Spherical geometry does not satisfy several of Euclid's axioms, including the parallel postulate.

For a spherical triangle, the sum of the angles is greater than 180° and can be up to 540°.

The amount by which the sum of the angles exceeds 180° is called the spherical excess, denoted as

of the triangle determine each other via the relation (called Girard's theorem):

: the spherical geometry locally resembles the euclidean one.

More generally, the euclidean law is recovered as a limit when the area tends to

More precisely, according to Lexell's theorem, given a spherical segment

Hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem.

A circle[5] cannot have arbitrarily small curvature,[6] so the three points property also fails.

Contrarily to the spherical case, the sum of the angles of a hyperbolic triangle is less than 180°, and can be arbitrarily close to 0°.

This relation was first proven by Johann Heinrich Lambert.

Once again, the euclidean law is recovered as a limit when the side lengths (or, more generally, the area) tend to

One can regard this limit as the case of ideal triangles, joining three points at infinity by three bi-infinite geodesics.

Lexell's theorem also has a hyperbolic counterpart: instead of circles, the level sets become pairs of curves called hypercycles, and the foliation is non-singular.

[8] In Taxicab Geometry, a type of non-Euclidean geometry where distance is measured using the Manhattan metric (only horizontal and vertical moves are allowed, like a grid), the concept of angle sum in a triangle becomes ambiguous.

In some interpretations, the sum of angles in a taxicab triangle can still be 180°, but the way angles are measured differs from Euclidean space.

Right angles can stretch or contract depending on the definition used, making the sum of angles a more flexible concept than in standard Euclidean geometry.

This discrepancy arises because, in taxicab geometry, the shortest path between two points is not necessarily a straight line in the Euclidean sense but rather a series of horizontal and vertical segments.

As a result, the definition of angles depends on the chosen metric, leading to alternative ways of measuring them.

For example, in some interpretations, a "right angle" may still resemble the familiar 90° turn, while in others, it may stretch depending on the path taken.

This flexibility in angle measurement makes taxicab geometry a fascinating field of study, particularly in urban planning, computer science, and optimization problems, where grid-based movement is common.

Exterior angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem.

In the differential geometry of surfaces, the question of a triangle's angular defect is understood as a special case of the Gauss-Bonnet theorem where the curvature of a closed curve is not a function, but a measure with the support in exactly three points – vertices of a triangle.