# Angle

An angle formed by two rays emanating from a vertex.

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1]Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

The word angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".[2]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.[3]

## Identifying angles

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is typically not used for this purpose.) Lower case Roman letters (abc, . . . ) are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples.

In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC (in Unicode U+2220 ANGLE) or ${\displaystyle {\widehat {\rm {BAC}}}}$. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").

Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.

Other Languages
Адыгэбзэ: Блыгу
Afrikaans: Hoek (meetkunde)
Alemannisch: Winkel (Geometrie)
العربية: زاوية (هندسة)
aragonés: Anglo
asturianu: Ángulu
Avañe'ẽ: Takamby
Aymar aru: K'uchu
azərbaycanca: Bucaq
تۆرکجه: آچی
বাংলা: কোণ
башҡортса: Мөйөш
беларуская: Вугал
беларуская (тарашкевіца)‎: Кут
български: Ъгъл
bosanski: Ugao
буряад: Үнсэг
català: Angle
čeština: Úhel
chiShona: Gonyo
Cymraeg: Ongl
dansk: Vinkel
Deutsch: Winkel
eesti: Nurk
Ελληνικά: Γωνία
español: Ángulo
Esperanto: Angulo
فارسی: زاویه
français: Angle
galego: Ángulo

한국어: 각 (수학)
հայերեն: Անկյուն
हिन्दी: कोण
hrvatski: Kut
Ido: Angulo
Ilokano: Anggulo
Bahasa Indonesia: Sudut (geometri)
interlingua: Angulo
italiano: Angolo
עברית: זווית
ქართული: კუთხე
kernowek: Angle
Kreyòl ayisyen: Ang
kurdî: Hoke
Кыргызча: Бурч
Latina: Angulus
latviešu: Leņķis
lietuvių: Kampas
Limburgs: Hook
lingála: Litúmu
magyar: Szög
македонски: Агол
മലയാളം: കോൺ
मराठी: कोन
Bahasa Melayu: Sudut
Nederlands: Hoek (meetkunde)
नेपाली: कोण
नेपाल भाषा: कुं

Nordfriisk: Winkel
norsk: Vinkel
norsk nynorsk: Vinkel
occitan: Angle
олык марий: Лук
oʻzbekcha/ўзбекча: Burchak
ਪੰਜਾਬੀ: ਕੋਨ
پنجابی: کونا
Patois: Hanggl
ភាសាខ្មែរ: មុំ
polski: Kąt
português: Ângulo
română: Unghi
Runa Simi: Chhuka
русский: Угол
Scots: Angle
sicilianu: Ànculu
සිංහල: කෝණය
Simple English: Angle
slovenčina: Uhol
slovenščina: Kot
Soomaaliga: Xagal
کوردی: گۆشە
српски / srpski: Угао
srpskohrvatski / српскохрватски: Ugao
Basa Sunda: Juru (élmu ukur)
suomi: Kulma
svenska: Vinkel
Tagalog: Anggulo
தமிழ்: கோணம்
татарча/tatarça: Почмак
తెలుగు: కోణం
ไทย: มุม
Türkçe: Açı
українська: Кут
اردو: زاویہ
Tiếng Việt: Góc

Winaray: Anggulo

ייִדיש: ווינקל