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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
, an angle is the figure formed by two rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
'' of the angle. Angles formed by two rays lie in the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
that contains the rays. Angles are also formed by the intersection of two planes. These are called
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, ele ...

dihedral angle
s. Two intersecting
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
s define also an angle, which is the angle of the
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
s at the intersection point. For example, the
spherical angle A spherical angle is a particular dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natu ...
formed by two
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
s on a
sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a circle in two-dimensional space. A sphere is the Locus (mathematics), set of points that are ...

sphere
equals the dihedral angle between the planes containing the great circles. ''Angle'' is also used to designate the measure of an angle or of a
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
. This measure is the ratio of the length of a
circular arc Circular may refer to: * The shape of a circle * Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fal ...
to its
radius In classical geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

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.


History and etymology

The word ''angle'' comes from the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

Latin
word ''angulus'', meaning "corner";
cognate In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most langu ...
words are the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
''(ankylοs)'', meaning "crooked, curved," and the
English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

English
word "
ankle The ankle, or the talocrural region, is the region where the foot The foot (plural: feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a Limb (anatomy), limb which bears weight and allows Animal locomoti ...

ankle
". Both are connected with the
Proto-Indo-European Proto-Indo-European (PIE) is the theorized common ancestor of the Indo-European language family The Indo-European languages are a language family A language is a structured system of communication used by humans, including speech ( ...
root ''*ank-'', meaning "to bend" or "bow".
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

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 Proclus Lycius (; 410/411/ 7 Feb. or 8 Feb. 412 –17 April 485 AD), called Proclus the Successor, Proclus the Platonic Successor, or Proclus of Athens (Greek: Προκλου Διαδοχου ''Próklos Diádochos'', ''"''in some Manuscript ...
, 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 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

straight line
; the second by
Carpus of Antioch In human anatomy The human body is the structure of a human being Humans (''Homo sapiens'') are the most populous and widespread species In biology, a species is the basic unit of biological classification, classification and ...
, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept.


Identifying angles

In
mathematical expressions In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constant (mathematics ...
, it is common to use
Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, Indo-European family of languages, nat ...
s (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angle (to avoid confusion with its other meaning, the symbol is typically not used for this purpose). Lower case Roman letters (''a'', ''b'', ''c'', . . . ) are also used, as are upper case Roman letters in the context of
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
s. 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 ) or \widehat. Where there is no risk of confusion, the angle may sometimes be referred to simply by its vertex (in this case "angle A"). Potentially, an angle denoted as, 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 Positive is a property of positivity and may refer to: Mathematics and science * Positive formulaIn mathematical logic, positive set theory is the name for a class of alternative set theory, set theories in which the axiom of comprehension *" ...
). However, in many geometrical situations, it is obvious from context that the positive angle less than or equal to 180 degrees is meant, in which case 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 the anticlockwise (positive) angle from C to B.


Types of angles


Individual angles

There is some common terminology for angles, whose measure is always non-negative (see #Positive and negative angles): * An angle equal to 0° or not turned is called a zero angle. * Angles smaller than a right angle (less than 90°) are called ''acute angles'' ("acute" meaning "sharp"). * An angle equal to turn (90° or radians) is called a ''
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

right angle
''. Two lines that form a right angle are said to be '' normal'', ''
orthogonal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

orthogonal
'', or ''
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
''. * Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called ''obtuse angles'' ("obtuse" meaning "blunt"). * An angle equal to turn (180° or radians) is called a ''straight angle''. * Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called ''reflex angles''. * An angle equal to 1 turn (360° or 2 radians) is called a ''full angle'', ''complete angle'', ''round angle'' or a ''perigon''. * Angles that are not right angles or a multiple of a right angle are called ''oblique angles''. The names, intervals, and measuring units are shown in the table below:


Equivalence angle pairs

* Angles that have the same measure (i.e. the same magnitude) are said to be ''equal'' or ''
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
''. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all ''right angles'' are equal in measure). * Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called ''coterminal angles''. * A ''reference angle'' is the acute version of any angle determined by repeatedly subtracting or adding straight angle ( turn, 180°, or radians), to the results as necessary, until the magnitude of the result is an acute angle, a value between 0 and turn, 90°, or radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180–150). An angle of 750 degrees has a reference angle of 30 degrees (750–720).


Vertical and adjacent angle pairs

When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other. * A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called ''vertical angles'' or ''opposite angles'' or ''vertically opposite angles''. They are abbreviated as ''vert. opp. ∠s''. :The equality of vertically opposite angles is called the ''vertical angle theorem''.
Eudemus of Rhodes Eudemus of Rhodes ( grc-gre, Εὔδημος) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly d ...
attributed the proof to
Thales of Miletus Thales of Miletus ( ; el, Θαλῆς Thales of Miletus ( ; el, Θαλῆς (ὁ Μιλήσιος), ''Thalēs''; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (fr ...

Thales of Miletus
. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: :* All straight angles are equal. :* Equals added to equals are equal. :* Equals subtracted from equals are equal. :When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle ''A'' equals ''x'', then the measure of angle ''C'' would be 180 − ''x''. Similarly, the measure of angle ''D'' would be 180 − ''x''. Both angle ''C'' and angle ''D'' have measures equal to 180 − ''x'' and are congruent. Since angle ''B'' is supplementary to both angles ''C'' and ''D'', either of these angle measures may be used to determine the measure of Angle ''B''. Using the measure of either angle ''C'' or angle ''D,'' we find the measure of angle ''B'' to be 180 − (180 − ''x'') = 180 − 180 + ''x'' = ''x''. Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure. * ''Adjacent angles'', often abbreviated as ''adj. ∠s'', are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called ''complementary'', ''supplementary'' and ''explementary'' angles (see "Combine angle pairs" below). A transversal is a line that intersects a pair of (often parallel) lines, and is associated with ''alternate interior angles'', ''corresponding angles'', ''interior angles'', and ''exterior angles''.


Combining angle pairs

Three special angle pairs involve the summation of angles: * ''Complementary angles'' are angle pairs whose measures sum to one right angle ( turn, 90°, or radians). If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
is 180 degrees, and the right angle itself accounts for 90 degrees. :The adjective complementary is from Latin ''complementum'', associated with the verb ''complere'', "to fill up". An acute angle is "filled up" by its complement to form a right angle. :The difference between an angle and a right angle is termed the ''complement'' of the angle. :If angles ''A'' and ''B'' are complementary, the following relationships hold: :: \begin & \sin^2A + \sin^2B = 1 & & \cos^2A + \cos^2B = 1 \\
pt
pt
& \tan A = \cot B & & \sec A = \csc B \end :(The
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
of an angle equals the
cotangent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

cotangent
of its complement and its secant equals the
cosecant In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

cosecant
of its complement.) :The
prefix A prefix is an affix In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) ...
" co-" in the names of some trigonometric ratios refers to the word "complementary". * Two angles that sum to a straight angle ( turn, 180°, or radians) are called ''supplementary angles''. :If the two supplementary angles are
adjacent Adjacent or adjacency may refer to: *Adjacent (graph theory), two vertices that are the endpoints of an edge in a graph *Adjacent (music), a conjunct step to a note which is next in the scale See also

*Adjacent angles, two angles that share ...
(i.e. have a common
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
and share just one side), their non-shared sides form a
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

straight line
. Such angles are called a ''linear pair of angles''. However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
are supplementary, and opposite angles of a
cyclic quadrilateral In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's metho ...

cyclic quadrilateral
(one whose vertices all fall on a single circle) are supplementary. :If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. :The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. :In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle. * Two angles that sum to a complete angle (1 turn, 360°, or 2 radians) are called ''explementary angles'' or ''conjugate angles''. *: The difference between an angle and a complete angle is termed the ''explement'' of the angle or ''conjugate'' of an angle.


Polygon-related angles

* An angle that is part of a
simple polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

simple polygon
is called an ''
interior angle 300px, Internal and external angles In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is Polygon#Convexity and non-convexity, con ...
'' if it lies on the inside of that simple polygon. A simple
concave polygon A simple polygon that is not convex polygon, convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex angle, reflex interior angle—that is, an angle with a measure that is between 180 degrees and 3 ...
has at least one interior angle that is a reflex angle. *: In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
, the measures of the interior angles of a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
add up to radians, 180°, or turn; the measures of the interior angles of a simple
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and ...

quadrilateral
add up to 2 radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
with ''n'' sides add up to (''n'' − 2) radians, or 180(''n'' − 2) degrees, (2''n'' − 4) right angles, or ( − 1) turn. * The supplement of an interior angle is called an ''
exterior angle 300px, Internal and external angles In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is co ...
'', that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
of the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
(or
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
) to decide the sign of the exterior angle measure. *: In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a ''supplementary exterior angle''. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons. * In a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

triangle
, the
bisectors
bisectors
of two exterior angles and the bisector of the other interior angle are
concurrent Concurrency, concurrent, or concurrence may refer to: * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion In law, a concurring opinion is in certain legal systems a written opinion An opin ...
(meet at a single point).Johnson, Roger A. ''Advanced Euclidean Geometry'', Dover Publications, 2007. * In a triangle, three intersection points, each of an external angle bisector with the opposite
extended side 300px, Each of a triangle's excircles (orange) is tangent to one of the triangle's sides and to the other two extended sides. In plane geometry, an extended side or sideline of a polygon In geometry Geometry (from the grc, γεωμετ ...
, are
collinear In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
. * In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear. * Some authors use the name ''exterior angle'' of a simple polygon to simply mean the ''explement exterior angle'' (''not'' supplement!) of the interior angle. This conflicts with the above usage.


Plane-related angles

* The angle between two planes (such as two adjacent faces of a
polyhedron In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
) is called a ''
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, ele ...

dihedral angle
''. It may be defined as the acute angle between two lines normal to the planes. * The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.


Measuring angles

The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are said to be ''equal'' or ''congruent'' or ''equal in measure''. In some contexts, such as identifying a point on a circle or describing the ''orientation'' of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
are effectively equivalent. In other contexts, such as identifying a point on a
spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
spiral
curve or describing the ''cumulative rotation'' of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent. In order to measure an angle , a
circular arc Circular may refer to: * The shape of a circle * Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fal ...
centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in
radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

radian
s. The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor , where ''k'' is the measure of a complete turn in the chosen unit (for example 360 for
degrees Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
or 400 for
gradian In trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), math ...
s): : \theta = k \frac. The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio ''s''/''r'' is unaltered. (Proof. The formula above can be rewritten as One turn, for which units, corresponds to an arc equal in length to the circle's
circumference In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
, which is 2''r'', so . Substituting ''n'' for ''θ'' and 2''r'' for ''s'' in the formula, results in )


Angle addition postulate

The angle addition postulate states that if ''B'' is in the interior of angle ''AOC'', then : m\angle AOC = m\angle AOB + m\angle BOC The measure of the angle ''AOC'' is the sum of the measure of angle AOB and the measure of angle ''BOC''. In this postulate it does not matter in which
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
the angle is measured as long as each angle is measured in the same unit.


Units

Units used to represent angles are listed below in descending magnitude order. Of these units, the ''
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
'' and the ''
radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

radian
'' are by far the most commonly used. Angles expressed in radians are dimensionless for
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric current) and units of measure ...
. Most units of angular measurement are defined such that one ''
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
'' (i.e. one full circle) is equal to ''n'' units, for some whole number ''n''. The two exceptions are the radian and the diameter part. ;
Turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
(''n'' = 1): The ''turn'', also ''cycle'', ''full circle'', ''revolution'', and ''rotation'', is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated , ''cyc'', ''rev'', or ''rot'' depending on the application, but in the acronym ''
rpm Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is the number of turns in one minute The minute is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the ...
'' (revolutions per minute), just ''r'' is used. A ''turn'' of ''n'' units is obtained by setting in the formula above. The equivalence of 1 ''turn'' is 360°, 2 rad, 400 grad, and 4 right angles. The symbol can also be used as a
mathematical constant A mathematical constant is a key whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an ), or by mathematicians' names to facilitate using it across multiple s. Constants arise in many areas of , with constan ...
to represent 2 radians. Used in this way () allows for radians to be expressed as a fraction of a turn. For example, half a turn is . ;
Quadrant Quadrant may refer to: Companies * Quadrant Cycle Company, 1899 manufacturers in Britain of the Quadrant motorcar * Quadrant (motorcycles), one of the earliest British motorcycle manufacturers, established in Birmingham in 1901 * Quadrant Private ...
(''n'' = 4): The ''quadrant'' is of a turn, i.e. a ''
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

right angle
''. It is the unit used in
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, ...
. 1 quad. = 90° =  rad = turn = 100 grad. In German the symbol has been used to denote a quadrant. ;
Sextant A sextant is a doubly reflecting navigation instrument that measures the angular distance Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle In Euclidean geometry, an angle is ...
(''n'' = 6): The ''sextant'' (''angle of the
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular polygon, equiangular; that is, all three internal angles are also con ...

equilateral triangle
'') is of a turn. It was the unit used by the
Babylonians Babylonia () was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia Mesopotamia ( ar, بِلَاد ٱلرَّافِدَيْن '; grc, Μεσοποταμία; Syriac language, Classical Syriac: ...
, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
subunits of the Babylonian unit. 1 Babylonian unit = 60° = /3 rad ≈ 1.047197551 rad. ;
Radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

Radian
(''n'' =  2 =  6.283 . . . ): The ''
radian The radian, denoted by the symbol \text, is the SI unit for measuring angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

radian
'' is the angle subtended by an arc of a circle that has the same length as the circle's radius. The case of radian for the formula given earlier, a ''radian'' of ''n'' = 2 units is obtained by setting ''k'' = = 1. One turn is 2 radians, and one radian is degrees, or about 57.2958 degrees. The radian is abbreviated ''rad'', though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

trigonometric function
s display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the system. ;
Clock position 300px, Points of a 12-hour clock A clock position, or clock bearing, is the direction of an object observed from a vehicle, typically a vessel or an aircraft, relative to the orientation of the vehicle to the observer. The vehicle must be con ...
(''n'' = 12): A clock position is the
relative direction Body relative directions (also known as egocentric coordinates) are orientation (geometry), geometrical orientations relative to a body such as a human person's. The most common ones are: left and right; forward(s) and backward(s); up and do ...
of an object described using the analogy of a
12-hour clock The 12-hour clock is a time convention in which the 24 hours of the day are divided into two periods: a.m. (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was ori ...
. One imagines a clock face lying either upright or flat in front of oneself, and identifies the twelve-hour markings with the directions in which they point. ;
Hour angle In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the direction of a point on the celestial sphere. The hour angle of a point is the angle between two planes: one contai ...
(''n'' = 24): The astronomical ''hour angle'' is of a turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called ''minute of time'' and ''second of time''. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° =  rad =  quad. = ''turn'' =  grad. ; (Compass) point or wind (''n'' = 32): The ''point'', used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...

navigation
, is of a turn. 1 point = of a right angle = 11.25° = 12.5 grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points. ; Hexacontade (''n'' = 60): The ''hexacontade'' is a unit of 6° that
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a ...

Eratosthenes
used, so that a whole turn was divided into 60 units. ; Pechus (''n'' = 144–180): The ''pechus'' was a Babylonian unit equal to about 2° or °. ; Binary degree (''n'' = 256): The ''binary degree'', also known as the ''
binary radian Binary scaling is a computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such a ...
'' (or ''brad''), is of a turn. The binary degree is used in computing so that an angle can be efficiently represented in a single
byte The byte is a unit of digital information that most commonly consists of eight bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It ...
(albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2''n'' equal parts for other values of ''n''. ;
Degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
(''n'' = 360): The ''degree'', denoted by a small superscript circle (°), is 1/360 of a turn, so one ''turn'' is 360°. The case of degrees for the formula given earlier, a ''degree'' of ''n'' = 360° units is obtained by setting ''k'' = . One advantage of this old
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for
geographical coordinates A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, spherical coordinate system using latitude, long ...
and in
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
and
ballistics Ballistics is the field of mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its ...
. ; Diameter part (''n'' = 376.99 . . . ): The ''diameter part'' (occasionally used in Islamic mathematics) is radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn. ; Grad (''n'' = 400):The ''grad'', also called ''grade'', ''
gradian In trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), math ...
'', or ''gon'', is of a turn, so a right angle is 100 grads. It is a decimal subunit of the quadrant. A
kilometre The kilometre (SI symbol: km; or ), spelt kilometer in American English, is a unit of length in the metric system, equal to one thousand metres (kilo- being the SI prefix for ). It is now the measurement unit used for expressing distances betw ...
was historically defined as a
centi ''Centi'' (symbol c) is a unit prefix A unit prefix is a specifier or mnemonic that is prepended to units of measurement to indicate multiples or fractions of the units. Units of various order of magnitude, sizes are commonly formed by the use o ...
-grad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
nautical mile. The grad is used mostly in
triangulation In trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...

triangulation
. ;
Milliradian A milliradian (International System of Units, SI-symbol mrad, sometimes also abbreviated mil) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of ...
: The milliradian (mil or mrad) is defined as a thousandth of a radian, which means that a rotation of one
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
consists of 2000π mil (or approximately 6283.185... mil), and almost all scope sights for
firearm A firearm is any type of gun A gun is a ranged weapon designed to use a shooting tube ( gun barrel) to launch typically solid projectiles, but can also project pressurized liquid (e.g. water guns/ cannons, spray guns for painting ...
s are calibrated to this definition. Also, there are three other derived definitions used for artillery and navigation which are ''approximately'' equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, which equals spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the true milliradian is approximately 0.05729578... degrees (3.43775... minutes). One "
NATO The North Atlantic Treaty Organization (NATO, ; french: Organisation du traité de l'Atlantique nord, ), also called the North Atlantic Alliance, is an intergovernmental organization, intergovernmental military alliance between 27 European ...
mil" is defined as of a circle. Just like with the true milliradian, each of the other definitions exploits the mil's handy property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away ( = 0.0009817... ≈ ). ;
Minute of arc A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of Angular unit, angular measurement equal to of one Degree (angle), degree. Since one degree is of a turn (geometry), turn (or complete rotatio ...
(''n'' = 21,600): The ''minute of arc'' (or ''MOA'', ''arcminute'', or just ''minute'') is of a degree = turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 +  = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 +  degrees. A
nautical mile A nautical mile is a unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country gl ...
was historically defined as a minute of arc along a
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
of the Earth. ;
Second of arc A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of Angular unit, angular measurement equal to of one Degree (angle), degree. Since one degree is of a turn (geometry), turn (or complete rotatio ...
(''n'' = 1,296,000): The ''second of arc'' (or ''arcsecond'', or just ''second'') is of a minute of arc and of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + + degrees, or 3.125 degrees. ;
Milliarcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol ', is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...
(''n'' = 1,296,000,000): mas ; Microarcsecond (''n'' = 1,296,000,000,000): µas


Positive and negative angles

Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent Orientation (geometry), orientations and/or Rotation (mathematics), rotations in opposite directions relative to some reference. In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The ''initial side'' is on the positive x-axis, while the other side or ''terminal side'' is defined by the measure from the initial side in radians, degrees, or turns. With ''positive angles'' representing rotations toward the positive y-axis and ''negative angles'' representing rotations toward the negative ''y''-axis. When Cartesian coordinates are represented by ''standard position'', defined by the ''x''-axis rightward and the ''y''-axis upward, positive rotations are anticlockwise and negative rotations are clockwise. In many contexts, an angle of −''θ'' is effectively equivalent to an angle of "one full turn minus ''θ''". For example, an orientation represented as  −45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of  −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a Vector (geometric), vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...

navigation
, bearing (navigation), bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.


Alternative ways of measuring the size of an angle

There are several alternatives to measuring the size of an angle by the angle of rotation. The ''grade (slope), grade of a slope'', or ''gradient'' is equal to the tangent (trigonometric function), tangent of the angle, or sometimes (rarely) the sine. A gradient is often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of the angle in radians. In rational geometry the ''spread (rational trigonometry), spread'' between two lines is defined as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.


Astronomical approximations

Astronomers measure angular separation of objects in degrees from their point of observation. * 0.5° is approximately the width of the sun or moon. * 1° is approximately the width of a little finger at arm's length. * 10° is approximately the width of a closed fist at arm's length. * 20° is approximately the width of a handspan at arm's length. These measurements clearly depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.


Angles between curves

The angle between a line and a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—''amphicyrtic'' (Gr. , on both sides, κυρτός, convex) or ''cissoidal'' (Gr. κισσός, ivy), biconvex; ''xystroidal'' or ''sistroidal'' (Gr. ξυστρίς, a tool for scraping), concavo-convex; ''amphicoelic'' (Gr. κοίλη, a hollow) or ''angulus lunularis'', biconcave.;


Bisecting and trisecting angles

The Greek mathematics, ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837 Pierre Wantzel showed that for most angles this construction cannot be performed.


Dot product and generalisations

In the Euclidean space, the angle ''θ'' between two Euclidean vectors u and v is related to their dot product and their lengths by the formula : \mathbf \cdot \mathbf = \cos(\theta) \left\, \mathbf \right\, \left\, \mathbf \right\, . This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.


Inner product

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product \langle \cdot , \cdot \rangle , i.e. : \langle \mathbf , \mathbf \rangle = \cos(\theta)\ \left\, \mathbf \right\, \ \left\, \mathbf \right\, . In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with : \operatorname \left( \langle \mathbf , \mathbf \rangle \right) = \cos(\theta)\ \left\, \mathbf \right\, \left\, \mathbf \right\, . or, more commonly, using the absolute value, with : \left, \langle \mathbf , \mathbf \rangle \ = , \cos(\theta) , \ \left\, \mathbf \right\, \ \left\, \mathbf \right\, . The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces \operatorname(\mathbf) and \operatorname(\mathbf) spanned by the vectors \mathbf and \mathbf correspondingly.


Angles between subspaces

The definition of the angle between one-dimensional subspaces \operatorname(\mathbf) and \operatorname(\mathbf) given by : \left, \langle \mathbf , \mathbf \rangle \ = , \cos(\theta), \left\, \mathbf \right\, \ \left\, \mathbf \right\, in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces \mathcal , \mathcal with \dim ( \mathcal) := k \leq \dim ( \mathcal) := l , this leads to a definition of k angles called canonical or principal angles between subspaces.


Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
s. Where ''U'' and ''V'' are tangent vectors and ''g''''ij'' are the components of the metric tensor ''G'', : \cos \theta = \frac.


Hyperbolic angle

A hyperbolic angle is an argument of a function, argument of a hyperbolic function just as the ''circular angle'' is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by Leonhard Euler in ''Introduction to the Analysis of the Infinite''.


Angles in geography and astronomy

In geography, the location of any point on the Earth can be identified using a ''geographic coordinate system''. This system specifies the latitude and longitude of any location in terms of angles subtended at the center of the Earth, using the equator and (usually) the Greenwich meridian as references. In
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several ''astronomical coordinate systems'', where the references vary according to the particular system. Astronomers measure the ''angular separation'' of two stars by imagining two lines through the center of the Earth, each intersecting one of the stars. The angle between those lines can be measured and is the angular separation between the two stars. In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude angle, altitude /elevation angle, elevation with respect to the horizon as well as the azimuth with respect to north. Astronomers also measure the ''apparent size'' of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.


See also

* Bisection#Angle bisector, Angle bisector * Angular velocity * Argument (complex analysis) * Astrological aspect * Central angle * Clock angle problem * Dihedral angle * Exterior angle theorem * Golden angle * Great circle distance * Inscribed angle * Irrational angle * Phase (waves) * Protractor * Solid angle for a concept of angle in three dimensions. * Spherical angle * Transcendent angle * Trisection * Zenith angle


Notes


References


Bibliography

* * . * * * * *


External links

* {{Authority control Angle,