Angle

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A green angle formed by two red rays on the Cartesian coordinate system

In Euclidean geometry, an angle or plane angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.<ref>Template:Harvnb</ref> Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection. Angles are also formed by the intersection of two planes; these are called dihedral angles. In any case, the resulting angle lies in a plane (spanned by the two rays or perpendicular to the line of plane-plane intersection).

The magnitude of an angle is called an angular measure or simply "angle". This measure, for an ordinary angle, is often visualized or defined using the arc of a circle centered at the vertex and lying between the sides. Two different angles may have the same measure, as in an isosceles triangle. "Angle" also denotes the angular sector, the infinite region of the plane bounded by the sides of an angle.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>Template:Efn

Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius, and may be a negative number; the arc is centered at the center of the rotation and delimited by any other point and its image after the rotation.

History and etymology

The word angle comes from the Latin word Template:Lang, meaning "corner". Cognate words include the Greek Template:Lang (Template:Lang) meaning "crooked, curved" and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".<ref>Template:Harvnb</ref>

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes, who regarded an angle as a deviation from a straight line; the second, angle as quantity, by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.<ref>Template:Harvnb; Template:Harvnb</ref>

Identifying angles

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angleTemplate:Sfn (the symbol Template:Math is typically not used for this purpose to avoid confusion with the constant denoted by that symbol). 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.

The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted Template:Math or <math>\widehat{\rm BAC}</math>. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A").

In other ways, an angle denoted as, say, Template:Math might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see Template:Section link). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, Template:Math always refers to the anticlockwise (positive) angle from B to C about A and Template:Math the anticlockwise (positive) angle from C to B about A.

TypesTemplate:Anchor

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Individual angles

There is some common terminology for angles, whose measure is always non-negative (see Template:Section link):

An angle equal to 0° or not turned is called a zero angle.Template:Sfn
An angle smaller than a right angle (less than 90°) is called an acute angleTemplate:Sfn ("acute" meaning "sharp").
An angle equal to Template:Sfrac turn (90° or Template:Sfrac radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.Template:Sfn
An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angleTemplate:Sfn ("obtuse" meaning "blunt").
An angle equal to Template:Sfrac turn (180° or Template:Math radians) is called a straight angle.Template:Sfn
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 2Template:Math 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:

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Unit	Interval
Name	zero angle	acute angle	right angle	obtuse angle	straight angle	reflex angle	perigon
turn	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap
radian	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap	Template:Nowrap
degree	0°	(0, 90)°	90°	(90, 180)°	180°	(180, 360)°	360°
gon	0^g	(0, 100)^g	100^g	(100, 200)^g	200^g	(200, 400)^g	400^g

Vertical and Template:Vanchor angle pairs

File:Vertical Angles.svg

Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. Hatch marks are used here to show angle equality.

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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. Template:Bulleted list A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.Template:Sfn

Combining angle pairs

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

<math display="block"> m\angle \mathrm{AOC} = m\angle \mathrm{AOB} + m\angle \mathrm{BOC} </math>

I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

Three special angle pairs involve the summation of angles: Template:Anchor

File:Complement angle.svg

The complementary angles a and b (b is the complement of a, and a is the complement of b.)

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Polygon-related angles

File:ExternalAngles.svg

Internal and external angles

An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle, that is, a reflex angle. Template:Pb In Euclidean geometry, the measures of the interior angles of a triangle add up to Template:Math radians, 180°, or Template:Sfrac turn; the measures of the interior angles of a simple convex quadrilateral add up to 2Template:Math radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)Template:Math radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)Template:Sfrac 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 must make at a vertex to trace the polygon.Template:Sfn 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. Still, one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure. Template:Pb 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, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).<ref name=Johnson>Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007.</ref>Template:Rp
In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.<ref name=Johnson/>Template:Rp
In a triangle, three intersection points, two 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.<ref name=Johnson/>Template:Rp
Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.<ref>Template:Citation as cited in Template:MathWorld</ref> This conflicts with the above usage.

Plane-related angles

The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle.<ref name="Chisholm 1911"/> 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 complementary to the angle between the intersecting line and the normal to the plane.

Measuring anglesTemplate:Anchor

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Measurement of angles is intrinsically linked with circles and rotation. The angle is first considered as within a circle of given size, centred at the vertex, and from that arrangement a ratio of certain lengths can be used quantify the angle size. As there are a number of valid measurement approaches, some have argued it useful to consider angle size as being proportional to these ratios, rather than defined by or equal to them.<ref>Template:Cite web</ref>

The usual characterization is to consider the smallest rotation of one of the rays about the vertex that maps it onto the other. The length of the arc, s, traced along the circle as the ray is rotated is said to be the arc length that is subtended by (or equivalently, subtends) the angle. However as s is dependent on the arbitrary choice of circle size, the ratio of length s to either the radius or circumference of the circle gives a general measure of angle size.

The ratio of the length s by the radius r is the number of radians in the angle, while the ratio of length s by the circumference C is the number of turns:<ref name="SIBrochure9thEd">Template:Citation</ref> <math display="block"> \theta = \frac{s}{r} \, \mathrm{rad}. </math><math display="block"> \theta = \frac{s}{ C} \ = \frac{s}{2\pi r} \, \mathrm{turns}. </math>

File:Angle measure.svg

The measure of angle Template:Math is Template:Nowrap.

The value of Template:Math 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.Template:Refn

Conventionally, in mathematics and the SI, the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted.

Other angular units are typically based on subdivisions of the turn and may then be obtained by multiplying the angle by a suitable conversion constant of the form Template:Sfrac, where k is the measure of a complete turn expressed in the chosen unit (for example, Template:Nowrap for degrees or 400 grad for gradians):

<math display="block"> \theta = \frac{k}{2\pi} \cdot \frac{s}{r}. </math>Angles of the same size are said to be equal 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 curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

Units

File:Angle radian.svg

Definition of 1 radian

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 (rad), and the gradian (grad), though many others have been used throughout history.<ref>Template:Cite web</ref>

In the International System of Quantities, an 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.

The following table lists some units used to represent angles.

Name	Number in one turn	In degrees	Description
radian	Template:Math	≈57°17′45″	The radian is determined by the circumference of a circle that is equal in length to the radius of the circle (n = 2Template:Pi = 6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is rad. One turn is 2Template:Math radians, and one radian is Template:Sfrac, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit rad being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI.
degree	360	1°	The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for geographical coordinates and in astronomy and ballistics (n = 360)
arcminute	21,600	0°1′	The minute of arc (or MOA, arcminute, or just minute) is Template:Sfrac of a degree = Template:Sfrac turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + Template:Sfrac = 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 + Template:Sfrac degrees. A nautical mile was historically defined as an arcminute along a great circle of the Earth. (n = 21,600).
arcsecond	1,296,000	0°0′1″	The second of arc (or arcsecond, or just second) is Template:Sfrac of a minute of arc and Template:Sfrac of a degree (n = 1,296,000). It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + Template:Sfrac + Template:Sfrac degrees, or 3.125 degrees. The arcsecond is the angle used to measure a parsec
grad	400	0°54′	The grad, also called grade, gradian, or gon. It is a decimal subunit of the quadrant. A right angle is 100 grads. A kilometre was historically defined as a centi-grad of arc along a meridian of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile (n = 400). The grad is used mostly in triangulation and continental surveying.
turn	1	360°	The turn is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2Template:Pi or [[Turn_(angle)#Proposals_for_a_single_letter_to_represent_2π\|Template:Tau (tau)]] radians.
hour angle	24	15°	The astronomical hour angle is Template:Sfrac turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = Template:Sfrac rad = Template:Sfrac quad = Template:Sfrac turn = Template:Sfrac grad.
(compass) point	32	11°15′	The point or wind, used in navigation, is Template:Sfrac of a turn. 1 point = Template:Sfrac of a right angle = 11.25° = 12.5 grad. Each point is subdivided into four quarter points, so one turn equals 128.
milliradian	Template:Math	≈0.057°	The true milliradian is defined as a thousandth of a radian, which means that a rotation of one turn would equal exactly 2000π mrad (or approximately 6283.185 mrad). Almost all scope sights for firearms are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which are approximately equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "NATO mil" is defined as Template:Sfrac of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away (Template:Sfrac = 0.0009817... ≈ Template:Sfrac).
binary degree	256	1°33′45″	The binary degree, also known as the binary radian or brad or binary angular measurement (BAM).<ref name="ooPIC"/> The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2ⁿ equal parts for other values of n. <ref name="Hargreaves_2010"/> It is Template:Sfrac of a turn.<ref name="ooPIC"/>
Template:Anchor Template:Pi radian	2	180°	The multiples of Template:Pi radians (MULTemplate:Pi) unit is implemented in the RPN scientific calculator WP 43S.<ref name="Bonin_2016"/> See also: IEEE 754 recommended operations
quadrant	4	90°	One quadrant is a Template:Sfrac turn and also known as a right angle. The quadrant is the unit in Euclid's Elements. In German, the symbol ^∟ has been used to denote a quadrant. 1 quad = 90° = Template:Sfrac rad = Template:Sfrac turn = 100 grad.
sextant	6	60°	The sextant was the unit used by the Babylonians,<ref name="Jeans_1947"/><ref name="Murnaghan_1946"/> The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the angle of the equilateral triangle or is Template:Sfrac turn. 1 Babylonian unit = 60° = Template:Pi/3 rad ≈ 1.047197551 rad.
hexacontade	60	6°	The hexacontade is a unit used by Eratosthenes. It equals 6°, so a whole turn was divided into 60 hexacontades.
pechus	144 to 180	2° to 2°30′	The pechus was a Babylonian unit equal to about 2° or Template:Sfrac°.
diameter part	≈376.991	≈0.95493°	The diameter part (occasionally used in Islamic mathematics) is Template:Sfrac radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
zam	224	≈1.607°	In old Arabia, a turn was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a turn is 224 zam.

Dimensional analysis

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Signed angles

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File:Angles on the unit circle.svg

Measuring from the x-axis, angles on the unit circle count as positive in the counterclockwise direction, and negative in the clockwise direction.

It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference.

In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are anticlockwise, and negative cycles are clockwise.

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined 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 in terms of an orientation, which is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

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

Equivalent angles

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.
The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).<ref>Template:Cite web</ref><ref>Template:Cite book</ref> Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo Template:Sfrac turn, 180°, or Template:Math radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

Related quantities

For an angular unit, it is definitional that the angle addition postulate holds. Some quantities related to angles where the angle addition postulate does not hold include:

The slope or gradient is equal to the tangent 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, such as the sine of the angle.

Angles between curves

File:Curve angles.svg

The angle between the two curves at P is defined as the angle between the tangents A and B at P.

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. Template:Lang, 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.<ref>Template:Harvnb; Template:Harvnb</ref>

Bisecting and trisecting angles

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

Dot product and generalisationsTemplate:Anchor

In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula

<math display="block"> \mathbf{u} \cdot \mathbf{v} = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

Inner product

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product <math> \langle \cdot , \cdot \rangle </math>, i.e.

<math display="block"> \langle \mathbf{u} , \mathbf{v} \rangle = \cos(\theta)\ \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

<math display="block"> \operatorname{Re} \left( \langle \mathbf{u} , \mathbf{v} \rangle \right) = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

or, more commonly, using the absolute value, with

<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> spanned by the vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> correspondingly.

Angles between subspaces

The definition of the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> given by

<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| </math>

in a Hilbert space can be extended to subspaces of finite dimensions. Given two subspaces <math> \mathcal{U} </math>, <math> \mathcal{W} </math> with <math> \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l </math>, this leads to a definition of <math>k</math> angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G,

<math display="block"> \cos \theta = \frac{g_{ij} U^i V^j}{\sqrt{ \left| g_{ij} U^i U^j \right| \left| g_{ij} V^i V^j \right|}}. </math>

Hyperbolic angle

A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case.<ref>Robert Baldwin Hayward (1892) The Algebra of Coplanar Vectors and Trigonometry, chapter six</ref> Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by Leonhard Euler in Introduction to the Analysis of the Infinite (1748).

Angles in geography and astronomy

In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the center of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the center of the Earth, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.

In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.

Astronomers also measure objects' apparent size as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can convert such an angular measurement into a distance/size ratio.

Other astronomical approximations include:

0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth.
1° is the approximate width of the little finger 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 depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

Unit	Symbol	Degrees	Radians	Turns	Other
Hour	h	15°	Template:Frac rad	Template:Frac turn
Minute	m	0°15′	Template:Frac rad	Template:Frac turn	Template:Frac hour
Second	s	0°0′15″	Template:Frac rad	Template:Frac turn	Template:Frac minute

Notes

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References

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Bibliography

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External links

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