TheInfoList

OR:

In
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, 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 Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
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 scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up mat ...

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 (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

to the respective curves at their point of intersection. ''Angle'' is also used to designate the of an angle or of a
rotation Rotation, or spin, is the circular movement of an object around a ''axis of rotation, central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A t ...
. 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 * Circ ...
to its
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''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 belonging to the Italic languages, 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 ...

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 etymological ancestor in a common parent language. Because language change can have radical ...
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 word "
ankle The ankle, or the talocrural region, or the jumping bone (informal) is the area where the foot and the human leg, leg meet. The ankle includes three joints: the ankle joint proper or talocrural joint, the subtalar joint, and the inferior tibiofi ...

". 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 philosophy, Greek Neoplatonism, Neoplatonist Philosophy, philosopher, ...
, 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 Greeks, 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 ...
, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept.

# Identifying angles

In mathematical expressions, 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 ...
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 rays 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"). * 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'', ''
orthogonal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

'', or ''
perpendicular In elementary geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works i ...

''. * 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 ''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''. 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, wikt:Θαλῆς, Θαλῆς; ) was a Greeks, Greek Greek Mathematics, mathematician, astronomer, statesman, and Pre-Socratic philosophy, pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of ...

. 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. * ''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: * ''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: :: $\begin & \sin^2A + \sin^2B = 1 & & \cos^2A + \cos^2B = 1 \\$& \tan A = \cot B & & \sec A = \csc B \end :(The
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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". * 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 vertex and share just one side), their non-shared sides form 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, ...

. 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 polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram ar ...

are supplementary, and opposite angles of a
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertex (geometry), vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices ...

(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, 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'' 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 ...
has at least one interior angle that is a reflex angle. *: In
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 ...
, the measures of the interior 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 ...

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 Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
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 tw ...
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'', 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
plane Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
(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 Visual perception, sight ...
) 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 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 ...

, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geom ...
, are
collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
. * 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 space, three-dimensional shape with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. A ...
) is called a ''
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up mat ...

''. 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 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
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 * Circ ...
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 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 ...
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 gradians): :$\theta = \frac \cdot \frac.$ 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.

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. These are known as angular units, with the most contemporary units being the degree ( ° ), the
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 ...
history History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as we ...
. Most units of angular measurement are defined such that one turn (i.e. the angle subtended by the circumference of a circle at its centre) 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 Quantity, 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 syste ...
, 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 quantity, base quantities (such as length, mass, time, and electric current) and units of measur ...
. 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 Th ...
and/or
rotations Rotation, or spin, is the circular movement of an object around a ''axis of rotation, central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A t ...
in opposite directions relative to some reference. In a two-dimensional
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
, 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 (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
, 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 (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
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 Rotation, or spin, is the circular movement of an object around a ''axis of rotation, central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or co ...
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 s ...
. 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities ...
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 motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navi ...
, bearings or
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an Angle#Measuring angles, angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly ...
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 Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...
'' or ''gradient'' is equal to the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...
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'' 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, 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 between objects in degrees from their point of observation. * 0.5° is the approximate diameter of the
Sun The Sun is the 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 st ...
and of the
Moon The Moon is Earth's only natural satellite. It is the List of natural satellites, 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 ( ...
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 Anatomical terms of location#Hands and feet, ulnar and smallest digit of the human hand, and next to the ring finger. Etymology The word "p ...
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 language, 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 associat ...
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 equinox (celestial coordinates), March equinox to the (hour circle of the) point in questio ...
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 ...
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
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 ...

(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These top ...
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 Idealiz ...
, 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 geometry, geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wa ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 (mathematics), scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidea ...
and their lengths by the formula :$\mathbf \cdot \mathbf = \cos\left(\theta\right) \left\, \mathbf \right\, \left\, \mathbf \right\, .$ 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 Mathematical object, object such as a Line (geometry), line, Ray (geometry), ray, or Euclidean vector, vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is ...
s and between
skew lines In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges ...
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 space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
, we replace the Euclidean dot product ( · ) by the inner product $\langle \cdot , \cdot \rangle$, i.e. :$\langle \mathbf , \mathbf \rangle = \cos\left(\theta\right)\ \left\, \mathbf \right\, \left\, \mathbf \right\, .$ In a complex
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
, the expression for the cosine above may give non-real values, so it is replaced with :$\operatorname \left\left( \langle \mathbf , \mathbf \rangle \right\right) = \cos\left(\theta\right) \left\, \mathbf \right\, \left\, \mathbf \right\, .$ or, more commonly, using the absolute value, with :$\left, \langle \mathbf , \mathbf \rangle \ = \left, \cos\left(\theta\right) \ \left\, \mathbf \right\, \left\, \mathbf \right\, .$ The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces $\operatorname\left(\mathbf\right)$ and $\operatorname\left(\mathbf\right)$ spanned by the vectors $\mathbf$ and $\mathbf$ correspondingly.

## Angles between subspaces

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

## Angles in Riemannian geometry

In
Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniq ...
, the
metric tensor In the mathematical field of differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of d ...
is used to define the angle between two
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

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 In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
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 a conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
of a
hyperbolic function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
just as the ''circular angle'' is the argument of a
circular function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. The comparison can be visualized as the size of the openings of a
hyperbolic sector A hyperbolic sector is a region (mathematics), region of the Cartesian plane bounded by a hyperbola and two ray (geometry), rays from the origin to it. For example, the two points and on the Hyperbola#Rectangular hyperbola, rectangular hyperbo ...
and a
circular sector A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (mathematics), disk (a closed region bounded by a circle) enclosed by two radius, radii and an Arc (geometry), arc, where the smaller area (geo ...
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 or planar lamina, while ''surface area'' refers to the area of an o ...
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 addition, 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 generalizat ...
in their angle argument, the circular ones are just
alternating series In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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''.

# Angles in geography and astronomy

In
geography Geography (from Ancient Greek, 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, i ...
, the location of any point on the Earth can be identified using a ''
geographic coordinate system The geographic coordinate system (GCS) is a spherical coordinate system, spherical or ellipsoidal coordinates (geodesy), ellipsoidal coordinate system for measuring and communicating position (geometry), positions directly on the Earth as lati ...
''. This system specifies the
latitude In geography, latitude is a Geographic coordinate system, 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 ...
and
longitude Longitude (, ) is a geographic coordinate system, geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another Celestial navigation, celestial body. It is an angular measurement, usually exp ...
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 Hemisphere, Northern and Southern Hemisphere, Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the ...
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
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 ...
, a given point on the
celestial sphere In astronomy and navigation, the celestial sphere is an abstraction, abstract sphere that has an arbitrarily large radius and is concentric objects, concentric to Earth. All objects in the sky can be conceived as being projective geometry, proje ...
(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 Natural satellite, satellites, planets, stars, galaxy, galaxies, and other celestial objects relative to physical reference points available to a situated obse ...
'', 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 In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' ...
'' 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 list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...
, 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 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 (geography), 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 equipotential su ...
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
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an Angle#Measuring angles, angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly ...
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.

* Angle measuring instrument * Angular statistics (
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude (mathematics), magnitude and sign (mathematics), sign) of a gi ...
,
standard deviation In statistics, the standard Deviation (statistics), deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (al ...
) *
Angle bisector In geometry, bisection is the division of something into two equal or congruence (geometry), congruent parts, usually by a line (mathematics), line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segm ...
*
Angular acceleration In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular accelera ...
*
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 ...
*
Angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...
*
Argument (complex analysis) In mathematics (particularly in complex analysis), the argument of a complex number ''z'', denoted arg(''z''), is the angle between the positive real number, real Cartesian coordinate system, axis and the line joining the origin and ''z'', repre ...
*
Astrological aspect In astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positio ...
*
Central angle A central angle is an angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of ...
* Clock angle problem *
Decimal degrees Decimal degrees (DD) is a notation for expressing latitude and longitude Geographic coordinate system, geographic coordinates as decimal fractions of a degree (angle), degree. DD are used in many geographic information systems (GIS), web mapping a ...
*
Dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up mat ...
* Exterior angle theorem *
Golden angle In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two Arc (geometry), arcs such that the ratio of the length of the smaller arc to the ...
*
Great circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two Point (geometry), points on the surface of a sphere, measured along the surface of the sphere ( ...
*
Inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chord (geometry), chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. E ...
*
Irrational angle In the mathematical theory of dynamical systems, an irrational rotation is a function (mathematics), map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 where ''θ'' is an irrational number. Under the i ...
*
Phase (waves) In physics and mathematics, the phase of a periodic function F of some real number, real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a sca ...
*
Protractor A protractor is a measuring instrument, typically made of transparent plastic or glass, for measuring angles. Some protractors are simple half-discs or full circles. More advanced protractors, such as the #Bevel, bevel protractor, have one or ...
*
Solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
* Spherical angle * Transcendent angle * Trisection *
Zenith angle The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere In astronomy and navigation, the celestial sphere is an abstraction, abstract sphere that has an arbitrarily large radius and is concentri ...

* * . * * * * *