Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, an angle is the figure formed by two

rays Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gra ...

, called the '' sides'' of the angle, sharing a common endpoint, called the ''

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

'' of the angle. Angles formed by two rays lie in the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

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, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

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 (ge ...

s may also define an angle, which is the angle of the rays lying

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

to the respective curves at their point of intersection. ''Angle'' is also used to designate the

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

of an angle or of a

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

. This measure is the ratio of the length of a

circular arc Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...

to its

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

. 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 belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

word ''angulus'', meaning "corner";

cognate In historical linguistics, cognates or lexical cognates are sets of words in different languages that have been inherited in direct descent from an etymology, etymological ancestor in a proto-language, common parent language. Because language c ...

words are the

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

''(ankylοs)'', meaning "crooked, curved," and the

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...

word "

ankle The ankle, or the talocrural region, or the jumping bone (informal) is the area where the foot and the leg meet. The ankle includes three joints: the ankle joint proper or talocrural joint, the subtalar joint, and the inferior tibiofibular join ...

". Both are connected with the

Proto-Indo-European Proto-Indo-European (PIE) is the reconstructed common ancestor of the Indo-European language family. Its proposed features have been derived by linguistic reconstruction from documented Indo-European languages. No direct record of Proto-Indo-E ...

root ''*ank-'', meaning "to bend" or "bow".

Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

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 (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...

, 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 In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...

; the second by

Carpus of Antioch Carpus of Antioch ( el, Κάρπος) was an ancient Greek mathematician. It is not certain when he lived; he may have lived any time between the 2nd century BC and the 2nd century AD. He wrote on mechanics, astronomy, and geometry. Proclus quotes f ...

, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept.

Identifying angles

mathematical expressions In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, fun ...

, it is common to use

Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...

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. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. In geometric figures, angles may also be identified by the three points that define them. For example, the angle with vertex A formed by the

AB and AC (that is, the lines from point A to points B and C) is denoted or

\widehat

. Where there is no risk of confusion, the angle may sometimes be referred to by its vertex (in this case "angle A"). Potentially, an angle denoted as, say, , 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 '). 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 always refers to the anticlockwise (positive) angle from B to C, and 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 '): * An angle equal to 0° or not turned is called a zero angle. * An angle smaller than a right angle (less than 90°) is called an ''acute angle'' ("acute" meaning "

sharp Sharp or SHARP may refer to: Acronyms * SHARP (helmet ratings) (Safety Helmet Assessment and Rating Programme), a British motorcycle helmet safety rating scheme * Self Help Addiction Recovery Program, a charitable organisation founded in 19 ...

"). * An angle equal to turn (90° or radians) is called a ''

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

''. Two lines that form a right angle are said to be ''

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

'', ''

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

'', or ''

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

''. * An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an ''obtuse angle'' ("obtuse" meaning "blunt"). * An angle equal to turn (180° or radians) is called a ''straight angle''. * An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a ''reflex angle''. * An angle equal to 1 turn (360° or 2 radians) is called a ''full angle'', ''complete angle'', ''round angle'' or a ''perigon''. * An angle that is not a multiple of a right angle is called an ''oblique angle''. 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 mod ...

''. 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 philosopher, considered the first historian of science, who lived from c. 370 BCE until c. 300 BCE. He was one of Aristotle's most important pupils, editing his teacher's work and m ...

attributed the proof to

Thales of Miletus Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him ...

. 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 . Similarly, the measure of angle ''D'' would be . Both angle ''C'' and angle ''D'' have measures equal to 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 . Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure. Adjacentangles

* ''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 ' 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: Complement angle

* ''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 with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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: ::

& \tan A = \cot B & & \sec A = \csc B \end

:(The

of an angle equals the

cotangent 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 ...

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 ...

of its complement.) :The

prefix A prefix is an affix which is placed before the Word stem, stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy'' ...

" co-" in the names of some trigonometric ratios refers to the word "complementary". Angle obtuse acute straight

* Two angles that sum to a straight angle ( turn, 180°, or radians) are called ''supplementary angles''. :If the two supplementary angles are adjacent (i.e. have a common

and share just one side), their non-shared sides form a

. 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, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

are supplementary, and opposite angles of a

cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

(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. Reflex 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, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise ...

is called an ''

interior angle 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 convex or non-convex, this angle is called an interior angle (or ) if ...

'' if it lies on the inside of that simple polygon. A simple

concave polygon A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Polyg ...

has at least one interior angle that is a reflex angle. *: In

, the measures of the interior angles of a

add up to radians, 180°, or turn; the measures of the interior angles of a simple

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

add up to 2 radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

with ''n'' sides add up to (''n'' − 2) radians, or (''n'' − 2)180 degrees, (''n'' − 2)2 right angles, or (''n'' − 2) turn. * The supplement of an interior angle is called an ''

exterior angle 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 convex or non-convex, this angle is called an interior angle (or ) if ...

'', 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 de ...

of the

(or

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

) 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

, the bisectors of two exterior angles and the bisector of the other interior angle are

concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

(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 In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a side arises in various contexts. Triangle In an obtuse triangle, the altitudes from the acute angled vertices i ...

, are

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

. * 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 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, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

) is called a ''

''. It may be defined as the acute angle between two lines

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 are effectively equivalent. In other contexts, such as identifying a point on a

spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are: In order to measure an angle θ, a

centered at the vertex of the angle is drawn, e.g. with a pair of

compasses A compass, more accurately known as a pair of compasses, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, it can also be used as a tool to mark out distances, in particular, on maps. Compasses c ...

. The ratio of the length s of the arc by the radius r of the circle is the number of

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s in the angle. Conventionally, in mathematics and in the SI, the radian is treated as being equal to the

dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

value 1. The angle expressed another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form , where ''k'' is the measure of a complete turn expressed in the chosen unit (for example, for degrees or 400 grad for

gradian In trigonometry, the gradian, also known as the gon (from grc, γωνία, gōnía, angle), grad, or grade, is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degree ...

s): :

\theta = \frac \cdot \frac.

The value of 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.

Angle addition postulate

The angle addition postulate states that if B is in the interior of angle AOC, then :

m\angle \mathrm = m\angle \mathrm + m\angle \mathrm

The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

Units

Throughout history, angles have been measured in various

units 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), ...

. These are known as angular units, with the most contemporary units being the

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

( ° ), the

(rad), and the

(grad), though many others have been used throughout

history History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the History of writing#Inventions of writing, invention of writing systems is considered prehistory. "History" is an umbr ...

. Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to ''n'' units, for some whole number ''n''. Two exceptions are the radian (and its decimal submultiples) and the diameter part. In the

International System of Quantities The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlie ...

, angle is defined as a dimensionless quantity, and in particular the radian unit is dimensionless. This convention impacts how angles are treated in

dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...

. For a discussion see . The following table list some units used to represent angles.

Signed 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

orientations ''Orientations'' is a bimonthly print magazine published in Hong Kong and distributed worldwide since 1969. It is an authoritative source of information on the many and varied aspects of the arts of East and Southeast Asia, the Himalayas, the India ...

and/or rotations in opposite directions relative to some reference. In a two-dimensional

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, an angle is typically defined by its two sides, with its vertex at the origin. The ''initial side'' is on the positive

x-axis A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, 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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

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 Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

and negative rotations are

clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

. 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 Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

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, ...

, bearings or

azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...

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 an angle

For an angular unit, it is definitional that the angle addition postulate holds. Some angle measurements where the angle addition postulate does not hold include: * The ''

slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...

'' or ''gradient'' is equal to the

of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. * The ''

spread Spread may refer to: Places * Spread, West Virginia Arts, entertainment, and media * ''Spread'' (film), a 2009 film. * ''$pread'', a quarterly magazine by and for sex workers * "Spread", a song by OutKast from their 2003 album ''Speakerboxxx/T ...

'' between two lines is defined in rational geometry 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. * Although done rarely, one can report the direct results of

trigonometric functions 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 ...

, such as the

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

of the angle.

Astronomical approximations

Astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...

s measure apparent sizes of and

distances Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

between objects in degrees from their point of observation. * 0.5° is the approximate diameter of the

Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...

and of the

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

as viewed from Earth. * 1° is the approximate width of the

little finger The little finger, or pinkie, also known as the baby finger, fifth digit, or pinky finger, is the most ulnar and smallest digit of the human hand, and next to the ring finger. Etymology The word "pinkie" is derived from the Dutch word ''pink ...

at arm's length. * 10° is the approximate width of a closed fist at arm's length. * 20° is the approximate 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 In English, the phrase ''rule of thumb'' refers to an approximate method for doing something, based on practical experience rather than theory. This usage of the phrase can be traced back to the 17th century and has been associated with various t ...

approximations only. In

astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

right ascension Right ascension (abbreviated RA; symbol ) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the earth. When paired w ...

and

declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...

are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

Angles between curves

The angle between a line and a

(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the

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

ancient Greek mathematicians Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...

knew how to bisect an angle (divide it into two angles of equal measure) using only a

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

, but could only trisect certain angles. In 1837,

Pierre Wantzel Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel pr ...

showed that for most angles this construction cannot be performed.

Dot product and generalisations

In the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, the angle ''θ'' between two

Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...

s u and v is related to their

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

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 vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...

s and between

skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the sa ...

from their vector equations.

Inner product

To define angles in an abstract real

inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...

, 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

, 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 \ = \left,  \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 \ = \left,  \cos(\theta) \ \left\,  \mathbf \right\,  \left\,  \mathbf \right\,

in a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

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 The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalized to arbitrary dimension. This generalization was first discussed by Jordan. For any pair of flats in a Eucl ...

between subspaces.

Angles in Riemannian geometry

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

, the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

is used to define the angle between two

s. Where ''U'' and ''V'' are tangent vectors and ''g''_''ij'' are the components of the metric tensor ''G'', :

\cos \theta = \frac.

Hyperbolic angle

hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions ...

is an

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

of a

hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...

just as the ''circular angle'' is the argument of a

circular function 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 ...

. The comparison can be visualized as the size of the openings of a

hyperbolic sector A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and i ...

and a

circular sector A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the large ...

since the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

s 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 mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

in their angle argument, the circular ones are just

alternating series In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternatin ...

forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

in ''

Introduction to the Analysis of the Infinite ''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...

''.

Angles in geography and astronomy

geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and ...

, the location of any point on the Earth can be identified using a ''

geographic coordinate system The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...

''. This system specifies the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

and

longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

of any location in terms of angles subtended at the center of the Earth, using the

equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...

and (usually) the

Greenwich meridian The historic prime meridian or Greenwich meridian is a geographical reference line that passes through the Royal Observatory, Greenwich, Royal Observatory, Greenwich, in London, England. The modern IERS Reference Meridian widely used today ...

as references. In

, a given point on the

celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...

(that is, the apparent position of an astronomical object) can be identified using any of several ''

astronomical coordinate systems Astronomical coordinate systems are organized arrangements for specifying positions of satellites, planets, stars, galaxies, and other celestial objects relative to physical reference points available to a situated observer (e.g. the true horizo ...

'', where the references vary according to the particular system. Astronomers measure the ''

angular separation Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer. Angular distance appears in mathematics (in pa ...

'' of two

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s by imagining two lines through the center of the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

, 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 In Euclidean 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. Angles formed by two rays lie in the plane that contains the rays. Angles are ...

such as

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

elevation The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vert ...

with respect to the

horizon The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...

as well as the

with respect to

north North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating Direction (geometry), direction or geography. Etymology T ...

. Astronomers also measure the ''apparent size'' of objects as an

angular diameter The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...

. For example, the

full moon The full moon is the lunar phase when the Moon appears fully illuminated from Earth's perspective. This occurs when Earth is located between the Sun and the Moon (when the ecliptic coordinate system, ecliptic longitudes of the Sun and Moon opp ...

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 The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians: : \begin \sin \theta &\approx \theta \\ \cos \theta &\approx 1 - \ ...

can be used to convert such an angular measurement into a distance/size ratio.

Notes

References

Bibliography

* * . * * * * *

External links

* {{Authority control