# Dictionary Definition

angle

### Noun

1 the space between two lines or planes that
intersect; the inclination of one line to another; measured in
degrees or radians

2 a biased way of looking at or presenting
something [syn: slant]

3 a member of a Germanic people who conquered
England and merged with the Saxons and Jutes to become
Anglo-Saxons

### Verb

1 move or proceed at an angle; "he angled his way
into the room"

2 to incline or bend from a vertical position;
"She leaned over the banister" [syn: lean, tilt, tip, slant]

3 seek indirectly; "fish for compliments" [syn:
fish]

4 fish with a hook

5 present with a bias; "He biased his
presentation so as to please the share holders" [syn: slant, weight]

# User Contributed Dictionary

see Angle

## English

### Pronunciation

- Rhymes: -æŋɡəl

### Etymology

From angle, from angulus.### Noun

- A figure formed by
two rays which start from a
common point (a plane angle)
or by three planes that
intersect (a solid angle).
- the angle between lines A and B

- The measure of such
a figure. In the case of a plane angle, this is the ratio (or proportional to the ratio)
of the arc
length to the radius
of a section of a
circle cut by the two
rays, centered at their common point. In the case of a solid angle,
this is the ratio of the surface area
to the square of the
radius of the section of a sphere.
- The angle between lines A and B is π/4 radians, or 45 degrees.

- A corner where two
walls intersect.
- an angle of a building

- A change in direction.
- The horse took off at an angle.

- A viewpoint.
- Look at it from this angle.

- The focus of a news story, either in print or broadcasting.
- slang professional wrestling A storyline between two wrestlers, providing the background for and approach to a feud.
- A scheme; a means of
benefitting from a situation, usually hidden, possibly illegal.
- His angle is that he gets a percentage, but mostly in trade.

#### Synonyms

- sense corner corner
- sense change in direction swerve, tangent
- sense vertex -gon (as per hexagon)
- sense viewpoint opinion, perspective, point of view, slant, view, viewpoint

#### Derived terms

- acute angle
- acute-angled
- central angle
- complementary angle
- dihedral angle
- exterior angle
- interior angle
- oblique angle
- obtuse angle
- obtuse-angled
- opposite angle
- pentangle
- plane angle
- quadrangle
- rectangle
- reflex angle
- right angle
- round angle
- solid angle
- straight angle
- supplementary angle
- triangle
- vertical angle

#### Related terms

geometrical figure

- Catalan: angle
- Czech: úhel
- Danish: vinkel
- Dutch: hoek
- Esperanto: angulo
- Finnish: kulma
- French: angle
- German: Winkel
- Greek: γωνία
- Hebrew:
- Hungarian: szög
- Icelandic: horn
- Italian: angolo
- Japanese: 角 (かく, kaku), 角度 (かくど, kakudo)
- Korean: 각 (角, gak)
- Lithuanian: kampas
- Polish: kąt
- Portuguese: ângulo
- Russian: угол
- Romanian: unghi
- Serbian: ugao, kut
- Slovak: uhol
- Slovene: kot
- Spanish: ángulo
- Swedish: vinkel , plan vinkel , rymdvinkel (solid angle)
- Telugu: కోణం
- Vietnamese: gốc

measure of such a figure

- Catalan: angle
- Danish: vinkel
- Dutch: hoek
- Esperanto: angulo
- French: angle
- German: Winkel
- Greek: γωνία (ɣonía)
- Italian: angolo
- Japanese: 角度 (かくど, kakudo)
- Korean: 각 (角, gak)
- Lithuanian: kampas
- Polish: kąt
- Portuguese: ângulo
- Romanian: unghi
- Russian: угол
- Slovene: kot
- Spanish: ángulo
- Swedish: vinkel
- Telugu: కోణం

corner where two walls intersect

- Catalan: racó , cantonada
- Danish: hjørne
- Dutch: hoek
- Esperanto: angulo
- German: Ecke
- Greek: γωνία, κόγχη
- Hebrew: ,
- Icelandic: horn
- Japanese: 角 (かく, kaku), 隅 (すみ, sumi)
- Korean: 귀 (gwi), 모서리 (moseori)
- Polish: kąt, róg
- Portuguese: esquina
- Romanian: colt
- Russian угол (úgol)
- Spanish: esquina (as seen from the outside), rincón (as seen from the inside)
- Swedish: hörn , knut (as seen from the outside)

change in direction

viewpoint

- Catalan: punt de vista, angle
- Danish: vinkel, synsvinkel, synspunkt
- Dutch: invalshoek , standpunt
- Finnish: näkökulma, kulma, perspektiivi, kantti
- French: point de vue
- German: Blickwinkel
- Greek: γωνία, θέα
- Hebrew: זווית ראיה
- Icelandic: sjónarhorn
- Italian: punto di vista
- Japanese: 見地, 立場
- Polish: strona , punkt widzenia , kąt
- Portuguese: ângulo , ponto de vista
- Romanian: punct de vedere
- Russian точка зрения (tóčka zrénija)
- Spanish: punto de vista
- Swedish: synvinkel
- Telugu: దృక్కోణం

- ttbc Breton: korn , -ioù p
- ttbc Indonesian: sudut
- ttbc Interlingua: angulo (1,2,3,4,5)
- ttbc Irish: uillinn

### Verb

#### Translations

fish with hook and line

# Extensive Definition

In order to measure an angle θ, a circular arc
centered at the vertex of the angle is drawn, e.g. with a pair of
compasses.
The length of the arc s is then divided by the radius of the circle
r, and possibly multiplied by a scaling constant k (which depends
on the units of measurement that are chosen):

- \theta = \frac(k).

The value of θ thus defined is independent of the
size of the circle: if the length of the radius is changed then the
arc length changes in the same proportion, so the ratio s/r is
unaltered.

In many geometrical situations, angles that
differ by an exact multiple of a full circle are effectively
equivalent (it makes no difference how many times a line is rotated
through a full circle because it always ends up in the same place).
However, this is not always the case. For example, when tracing a
curve such as a spiral
using polar
coordinates, an extra full turn gives rise to a quite different
point on the curve.

### Units

Angles are considered dimensionless, since they
are defined as the ratio of lengths. There are, however, several
units used to measure angles, depending on the choice of the
constant k in the formula above. Of these units, treated in more
detail below, the degree and the radian are by far the most
common.

With the notable exception of the radian, most
units of angular measurement are defined such that one full circle
(i.e. one revolution) is equal to n units, for some whole number n.
For example, in the case of degrees, A full circle of n units is
obtained by setting in the formula above. (Proof. The formula above
can be rewritten as One full circle, for which units, corresponds
to an arc equal in length to the circle's circumference, which is
2πr, so . Substituting n for θ and 2πr for s in the formula,
results in )

- The degree,
denoted by a small superscript circle (°) is 1/360 of a full
circle, so one full circle 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. (The goal of having all "interesting"
angles measured as whole numbers is of course unattainable.)
Fractions of a degree may be written in normal decimal notation
(e.g. 3.5° for three and a half degrees), but the following
sexagesimal subunits of the "degree-minute-second" system are also
in use, especially for
geographical coordinates and in astronomy and ballistics:
- The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
- The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

- The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. 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 system.

- The mil is approximately equal to a milliradian. There are several definitions.

- The right angle is 1/4 of a full circle. It is the unit used in Euclid's Elements. 1 right angle = 90° = π/2 rad = 100 gon.

- The angle of the equilateral triangle is 1/6 of a full circle. It was the unit used by the Babylonians, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.
- The grad, also called grade, gradian, or gon is 1/400 of a full circle, so one full circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right angle. A kilometer was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.

- The point, used in navigation, is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is called "boxing the compass". 1 point = 1/8 of a right angle = 11.25° = 12.5 gon.

- The astronomical hour angle is 1/24 of a full circle. The sexagesimal subunits were called minute of time and second of time (even though they are units of angle). 1 hour = 15° = π/12 rad = 1/6 right angle ≈ 16.667 gon.

- The binary degree, also known as the binary radian (or brad), is 1/256 of a full circle. The binary degree is used in computing so that an angle can be efficiently represented in a single byte.

- The grade of a slope, or gradient, is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

### Positive and negative angles

A convention universally adopted in mathematical
writing is that angles given a sign are positive angles if measured
counterclockwise, and negative angles if measured
clockwise, from a given line. If no line is specified, it can
be assumed to be the x-axis in the
Cartesian
plane. In many geometrical situations a negative angle of
−θ is effectively equivalent to a positive angle of "one
full rotation less θ". For example, a clockwise rotation of 45°
(that is, an angle of −45°) is often effectively
equivalent to a counterclockwise rotation of
360° − 45° (that is, an angle of
315°).

In three dimensional geometry, "clockwise" and
"counterclockwise" have no absolute meaning, so the direction of
positive and negative angles must be defined relative to some
reference, which is typically a vector
passing through the angle's vertex and perpendicular to the plane
in which the rays of the angle lie.

In navigation, bearings
are measured from north, increasing clockwise, so a bearing of 45
degrees is north-east. Negative bearings are not used in
navigation, so north-west is 315 degrees.

### Approximations

- 1° is approximately the width of a little finger at arm's length
- 10° is approximately the width of a closed fist at arm's length.
- 20° is approximately the width of a handspan at arm's length.

## Identifying angles

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

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

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

## Types of angles

- An angle of 90° (π/2
radians, or one-quarter of the full circle) is called a right angle.
- Two lines that form a right angle are said to be perpendicular or orthogonal.

- Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
- Angles larger than a right angle and smaller than two right angles (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
- Angles equal to two right angles (180°) are called straight angles.
- Angles larger than two right angles but less than a full circle (between 180° and 360°) are called reflex angles.
- Angles that have the same measure are said to be congruent.
- Two angles opposite each other, formed by two intersecting straight lines that form an "X" like shape, are called vertical angles or opposite angles. These angles are congruent.
- Angles that share a common vertex and edge but do not share any interior points are called adjacent angles.
- Two angles that sum to one right angle (90°) are called
complementary
angles.
- The difference between an angle and a right angle is termed the complement of the angle.

- Two angles that sum to a straight angle (180°) are called
supplementary
angles.
- The difference between an angle and a straight angle is termed the supplement of the angle.

- Two angles that sum to one full circle (360°) are called explementary angles or conjugate angles.
- An angle that is part of a simple
polygon is called an interior
angle if it lies in the inside of that the simple polygon. Note
that in a simple polygon that is concave, at least one interior
angle exceeds 180°.
- In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, or 180°; the measures of the interior angles of a simple quadrilateral add up to 2π radians, or 360°. In general, the measures of the interior angles of a simple polygon with n sides add up to [(n − 2) × π] radians, or [(n − 2) × 180]°.

- The angle supplementary to the interior angle is called the
exterior
angle. It measures the amount of "turn" one has to make at this
vertex to trace out the polygon. If the corresponding interior
angle exceeds 180°, 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
of the plane
(or surface) to decide
the sign of the exterior angle measure.
- In Euclidean geometry, the sum of the exterior angles of a simple polygon will be 360°, one full turn.

- Some authors use the name exterior angle of a simple polygon to simply mean the explementary (not supplementary!) of the interior angle http://mathworld.wolfram.com/ExteriorAngle.html. This conflicts with the above usage.
- The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.
- The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.
- If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are congruent; adjacent angles are supplementary (that is, their measures add to π radians, or 180°).

## A formal definition

### Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if \theta is a Euclidean angle, it is true that- \cos \theta = \frac

and

- \sin \theta = \frac

for two numbers x and y. So an angle in the
Euclidean plane can be legitimately given by two numbers x and
y.

To the ratio \frac there correspond two angles in
the geometric range 0 , since

- \frac = \frac = \frac = \frac = \frac.

### Using rotations

Suppose we have two unit vectors \vec and \vec in the euclidean plane \mathbb^2. Then there exists one positive isometry (a rotation), and one only, from \mathbb^2 to \mathbb^2 that maps u onto v. Let r be such a rotation. Then the relation \vec\mathcal\vec defined by \vec=r(\vec) is an equivalence relation and we call angle of the rotation r the equivalence class \mathbb/\mathcal, where \mathbb denotes the unit circle of \mathbb^2. The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector (1,0), then for any point M on \mathbb at distance \theta from (1,0) (on the circle), let \vec=\overrightarrow. If we call r_\theta the rotation that transforms (1,0) into \vec, then \left[r_\theta\right]\mapsto\theta is a bijection, which means we can identify any angle with a number between 0 and 2\pi.## Angles between curves

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. μφί, 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.## The dot product and generalisation

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula- \mathbf \cdot \mathbf = \cos(\theta)\ \|\mathbf\|\ \|\mathbf\|.

This allows one to define angles in any real
inner
product space, replacing the Euclidean dot product · by the
Hilbert
space inner product .

## 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 gij are the components of the metric tensor G,\cos \theta = \frac .

## Angles in geography and astronomy

In geography we specify the
location of any point on the Earth using a
Geographic coordinate system. This system specifies the
latitude and longitude of any location, in
terms of angles subtended at the centre of the Earth, using the
equator and (usually)
the Greenwich
meridian as references.

In astronomy, we similarly
specify a given point on the celestial
sphere using any of several
Astronomical coordinate systems, where the references vary
according to the particular system.

Astronomers can also measure the angular
separation of two stars by
imagining two lines through the centre of the Earth, each
intersecting one of the stars. The angle between those lines can be
measured, and is the angular separation between the two
stars.

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

## References

- .
- Preliminary">http://www.utexas.edu/cola/centers/lrc/ielex/PokornyMaster-X.html.

## See also

- Complementary angles
- Supplementary angles
- Central angle
- Inscribed angle
- Solid angle for a concept of angle in three dimensions.
- Astrological aspect
- Protractor
- Clock angle problem

## External links

- Angle Bisectors in a Quadrilateral at cut-the-knot
- Constructing a triangle from its angle bisectors at cut-the-knot
- Convert angles in sexagesimal degree format to decimal degrees, and vice-versa
- Angle Estimation -- for basic astronomy.
- Angle definition pages with interactive applets.
- Various angle constructions with compass and straightedge Animated demonstrations

angle in Afrikaans: Hoek (meetkunde)

angle in Tosk Albanian: Winkel (Geometrie)

angle in Arabic: زاوية(هندسة)

angle in Asturian: Ángulu

angle in Guarani: Takamby

angle in Aymara: K'uchu

angle in Bengali: সমকোণ

angle in Bosnian: Ugao

angle in Bulgarian: Ъгъл

angle in Catalan: Angle

angle in Czech: Úhel

angle in Danish: Vinkel

angle in German: Winkel

angle in Modern Greek (1453-): Γωνία

angle in Spanish: Ángulo

angle in Esperanto: Angulo

angle in Persian: زاویه

angle in French: Angle

angle in Scottish Gaelic: Ceàrn
(Matamataig)

angle in Galician: Ángulo

angle in Classical Chinese: 角

angle in Korean: 각도

angle in Croatian: Kut

angle in Ido: Angulo

angle in Indonesian: Sudut (geometri)

angle in Icelandic: Bogagráða

angle in Italian: Angolo

angle in Hebrew: זווית

angle in Swahili (macrolanguage): Pembe
(jiometria)

angle in Haitian: Ang

angle in Latvian: Leņķis

angle in Lithuanian: Kampas

angle in Hungarian: Szög

angle in Macedonian: Агол

angle in Marathi: कोन

angle in Malay (macrolanguage): Sudut

angle in Dutch: Hoek (meetkunde)

angle in Japanese: 角度

angle in Polish: Kąt płaski

angle in Portuguese: Ângulo

angle in Romanian: Ângulo

angle in Quechua: Chhuka

angle in Russian: Угол

angle in Simple English: Angle

angle in Slovak: Uhol

angle in Slovenian: Kot

angle in Serbian: Угао

angle in Finnish: Kulma

angle in Swedish: Vinkel

angle in Tamil: கோணம்

angle in Thai: มุม

angle in Vietnamese: Góc

angle in Turkish: Açı

angle in Ukrainian: Кут

angle in Chinese: 角

# Synonyms, Antonyms and Related Words

Anschauung, L, action, aim, anagnorisis, angle for,
angle of vision, angle off, apex, approach, architectonics, architecture, argument, aspect, atmosphere, background, bait the hook,
base, basis, be after, bear off,
bend, bias, bifurcate, bifurcation, bight, bob, bow, branch, brew, cabal, cant, catastrophe, characterization,
chevron, clam, coign of vantage, coin, collude, color, complication, complot, concoct, configuration, connive, conspire, continuity, contrivance, contrive, cook up, corner, countermine, counterplot, crank, crook, crotchet, cusp, dap, deflect, deflection, denouement, design, detail, development, deviate, device, dib, dibble, direction, distance, divagate, diverge, dogleg, drive, edge, effect, eidolon, elbow, ell, engineer, episode, eye, fable, facet, falling action, fashion, feature, figure, finagle, finesse, fish, fish for, flection, flexure, fly-fish, footing, fork, form, frame, frame of reference, frame
up, framework,
furcate, furcation, gerrymander, gestalt, gig, gimmick, go fishing, grig, ground, guddle, guise, hand, hatch, hatch a plot, hatch up,
hook, hunt for, image, imago, impression, incident, inflection, intersection, intrigue, item, jack, jacklight, jig, jockey, knee, lay a plot, light, likeness, line, lineaments, local color,
look, look for, machinate, maneuver, manipulate, manner, mental outlook, mood, motif, movement, mythos, net, nook, oblique, oblique angle, operate, outlook, particular, peripeteia, perspective, phase, phasis, place, plan, play games, plot, point, point of view, position, post, projection, pull strings,
quoin, recognition, reference, reference system,
regard, respect, rig, right angle, rising action,
salient angle, scheme,
seat, secondary plot,
seek, seeming, seine, semblance, shape, sheer, shrimp, side, sight, simulacrum, situation, skew, slant, slue, spin, stand, standing, standpoint, station, status, still-fish, story, structure, style, subject, subplot, sway, swerve, switch, system, thematic development,
theme, tone, topic, torch, total effect, trawl, troll, try for, turn, turning, twist, universe, vantage, vantage point, veer, venue, vertex, view, viewpoint, wangle, whale, wise, zag, zig, zigzag