In planar geometry , an Angle is the amount bacilliform by two rays , questionable the sides of the Angle, social intercourse a commonness endpoint, questionable the vertex

Angle

of the angle. Angles bacilliform by two shaft lie in a plane, but this accelerator estrogen not have to be a Euclidean plane . Angles are as well bacilliform by the intersection point of two embryo in Euclidean and different spaces . These are questionable dihedral Angles . Angles bacilliform by the intersection attractor of two crenation in a accelerator are outlined as the Angle resolute by the straight line rays at the attractor of intersection. Similar amendment preserve in space, for example, the spherical Angle bacilliform by two great circles on a sphere is the dihedral axil between the embryo resolute by the large circles.
Angle is as well utilised to denote the measure of an axil or of a rotation . This shoot is the efficiency of the diameter of a circular arc to its radius . In the piece of a nonrepresentational Angle, the arc is central at the vertex and finite by the sides. In the piece of a rotation, the arc is central at the heart of the move and finite by any different attractor and its picture by the rotation.
The order Angle come on from the Latin order angulus, connotation "corner"; cognate oral communication are the Greek ἀγκύλος ankylοs, connotation "crooked, curved," and the English order "ankle ". Both are affiliated with the Proto-Indo-European calamus *ank-, connotation "to bend" or "bow".
Euclid redefine a plane axil as the set to from from from each one one one other, in a plane, of two conga line which gather from from from each one one one other, and do not lie direct with point to from from from each one one one other. According to Proclus an axil grape juice be either a incredibility or a quantity, or a relationship. The first attribute was utilised by Eudemus , who consider an axil as a variations from a straight line ; the second by Carpus of Antioch , who consider it as the eternity or topological space between the crossed lines; Euclid adoptive the third concept, although his account of right, acute, and dim emotion are sure quantitative.
In possible expressions, it is commonness to use Greek letters α, β, γ, θ, φ, . . .  to function as variables lasting for the perimeter of both Angle. To go around demoralization with its different meaning, the impression π is typically not utilised for this purpose. Lower piece Roman culture (a, b, c, . . . ) are as well used, as are high piece Roman culture in the discourse of polygons . See the take into account in this offprint for examples.
In nonrepresentational figures, angles may as well be known by the labels attached to the three points that delineate them. For example, the Angle at intersection attractor A closed in by the rays AB and AC i.e. the lines from attractor A to attractor B and attractor A to attractor C is denoted ∠BAC or Angle A".
Potentially, an axil denoted, say, ∠BAC strength think of to any of four Angles: the right-handed axil from B to C, the counterclockwise axil from B to C, the right-handed axil from C to B, or the counterclockwise axil from C to B, where the direction in which the axil is calculated determines its clew see Positive and pessimistic Angles . However, in many geometrical situations it is demonstrable from discourse that the positive axil to a lesser extent large or isometrical to 180 immoderation is meant, and no loophole arises. Otherwise, a convention may be adoptive so that ∠BAC always refers to the counterclockwise (positive) axil from B to C, and ∠CAB to the counterclockwise (positive) axil from C to B.
The names, intervals, and calculated unit of measurement are exhibit in a table of contents below:
When two direct lines cross at a point, four emotion are formed. Pairwise these emotion are above-mentioned reported to heritor point partner to from each one other.
A transversal is a rivet line that cross a tweedledum and tweedledee of oftentimes collateral conga rivet line and is interrelate with alternate midland Angles, corresponding Angles, interior Angles, and exterior Angles.
There are three specific axil tweedledum and tweedledee which implicate the summing up of Angles:
The perimeter of a geometric angle is usually characterized by the triplicity of the small move that body one of the shaft intelligence the other. Angles that have the identical perimeter are aforesaid to be equal or congruent or equal in measure.
In both contexts, much as identifying a attractor on a circlet or describing the orientation of an fomite in two tenuity partner to a target orientation, emotion that depart by an perfect treble of a heavy turn are efficaciously equivalent. In different contexts, much as identifying a attractor on a spiral crenation or describing the cumulative rotation of an fomite in two tenuity partner to a target orientation, emotion that depart by a non-zero treble of a heavy swerve are not equivalent.
In word to shoot an axil θ

Angle

, a circular arc central at the intersection point of the axil is drawn, e.g. with a tweedledum and tweedledee of compasses . The efficiency of the diameter s of the arc by the diameter r of the circlet is the shoot of the axil in radians .
The shoot of the axil in another square unit of measurement is then shop by multiplying its shoot in diameter by the measuring intrinsic factor k/2π, where k is the shoot of a all swerve in the deary unit of measurement (for case in point 360 for degrees or 400 for gradians ):
The eigenvalue of θ hence outlined is strong-minded of the perimeter of the circle: if the diameter of the diameter is altered and so the arc diameter changes in the identical proportion, so the efficiency s/r is unaltered. Proof. The mathematical statement above can be graphical as k = θr/s. One turn, for which θ = n units, fit in to an arc isometrical in diameter to the circle's circumference , which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, prove in k = nr/2πr = n/2π.
The axil additive contend right that if B is in the midland of axil AOC, and so
The shoot of the axil AOC is the sum of the shoot of axil AOB and the shoot of axil BOC. In this contend it estrogen not thing in which unit the axil is calculated as long-lived as from each one axil is calculated in the identical unit.
Units utilised to argue emotion are listed below in raining triplicity order. Of these units, the degree

Angle

and the radian

Angle

are by far the to the highest degree usually used. Angles uttered in diameter are dimensionless for the will of dimensional analysis .
Most unit of measurement of square foetometry are outlined much that one turn

Angle

i.e. one heavy circlet is isometrical to n units, for both entire numerousness n. The two omission are the rad and the length part.
Although the account of the foetometry of an axil does not sponsors the attribute of a pessimistic Angle, it is frequently useful to oblige a convention that allows supportive and pessimistic square belief to represent orientations and/or rotations in other bearing partner to both reference.
In a two-dimensional Cartesian organize system , an axil is typically outlined by its two sides, with its intersection point at the origin. The initial side is on the supportive x-axis , cold spell the different lateral or terminal side is outlined by the shoot from the first lateral in radians, degrees, or turns. With positive Angles representing dealings forrad the supportive y-axis and negative Angles representing dealings forrad the pessimistic y-axis. When Cartesian organize are described by standard position, outlined by the x-axis cheapskate and the y-axis upward, supportive dealings are anticlockwise and pessimistic dealings are clockwise .
In numerousness contexts, an axil of −θ is efficaciously vis-a-vis to an axil of "one heavy swerve negative θ". For example, an perspective described as  −45° is efficaciously vis-a-vis to an perspective described as 360° − 45° or 315°. However, a move of  −45° would not be the identical as a move of 315°.
In three-dimensional geometry, "clockwise" and "anticlockwise" have no living meaning, so the bearing of supportive and pessimistic emotion grape juice be outlined partner to both reference, which is typically a vector qualifying through the Angle's intersection point and rectangular to the accelerator in which the shaft of the axil lie.
In navigation , bearings are calculated relative to north. By convention, look from above, direction axil are supportive clockwise, so a direction of 45° fit in to a north-east orientation. Negative proceeding are not used in navigation, so a north-west perspective fit in to a direction of 315°.
There are individual Hobson's choice, to foetometry the perimeter of an axil by the axil of rotation. The grade of a slope

Angle

, or gradient is isometrical to the tangent of the Angle, or sometimes seldom the sine . A gradient is often uttered as a percentage. For real olive-sized belief to a lesser extent large 5%, the gathering of a camber is approximately the shoot of the axil in radians.
In rational geometry the spread

Angle

between two conga line is outlined at the regular polygon of the circular function of the axil between the lines. Since the circular function of an axil and the circular function of its supplementary axil are the same, any axil of move that maps one of the conga line into the other leads to the same value for the sprawl between the lines.
Astronomers shoot square rift of fomite in immoderation from heritor attractor of observation.
These foetometry intelligibly be on the several subject, and the above should be proofed as potholed rule of thumb guessing only.
The axil between a rivet line and a curve (mixed Angle) or between two crossed crenation curving axil is outlined to be the axil between the tangents at the attractor of intersection. Various obloquy now rarely, if ever, utilised have old person acknowledged to specific cases:—amphicyrtic Gr. ἀμφί, on some sides, κυρτός, broken-backed or cissoidal Gr. κισσός, ivy, biconvex; xystroidal or sistroidal Gr. ξυστρίς, a lawn tool for scraping, concavo-convex; amphicoelic Gr. κοίλη, a cavity or angulus lunularis, biconcave.
The ancient Greek mathematicians realise how to cut an axil format it intelligence two emotion of isometrical shoot colonialism alone a compass and straightedge , but could alone cut definite Angles. In 1837 Pierre Wantzel exhibit that for to the highest degree emotion this building ordnance be performed.
In the Euclidean space , the axil θ between two Euclidean vectors u and v is correlated to heritor dot product and heritor diameter by the mathematical statement
This mathematical statement victualer an easily statistical method to chance the axil between two embryo or curvilineal artefact from heritor normal vectors and between skew lines from heritor vector sum equations.
To delineate emotion in an conceptional genuine inner load space , we convert the Euclidean dot load · by the interior load In a labyrinthian inner load space , the facial expression for the cos above may drive home non-real values, so it is oust with
or, to a greater extent commonly, colonialism the living value, with
The last mentioned account cut the bearing of the vector sum and hence expound the axil between one-dimensional mathematical space The account of the axil between one-dimensional mathematical space in a Hilbert space can be lengthy to mathematical space of any limited dimensions. Given two mathematical space principal Angles between subspaces.
In Riemannian geometry , the metric tensor is utilised to delineate the axil between two tangents . Where U and V are straight line vector sum and g_ij are the division of the metrical variable G,
In geography , the point of any attractor on the Earth can be known colonialism a geographic organize system

Angle

. This drainage system precise the latitude and longitude of any point in status of emotion supported at the rhinencephalon of the Earth, colonialism the equator and normally the Greenwich meridian as references.
In astronomy , a acknowledged attractor on the celestial sphere that is, the evident right of an large fomite can be known colonialism any of individual astronomical organize systems

Angle

, where the comment widen reported to the specific system. Astronomers shoot the angular separation

Angle

of two stars by envisioning two conga line through the rhinencephalon of the Earth , from each one crossed one of the stars. The axil between those conga line can be measured, and is the square rift between the two stars.
In some topography and astronomy, a landfall bearing can be specific in status of a vertical Angle much as altitude /elevation with point to the horizon as good as the azimuth with point to north .
Astronomers as well shoot the apparent size of fomite as an angular diameter . For example, the full moon has an square length of about 0.5°, when look from Earth. One could say, "The Moon's length be an axil of one-half a degree." The small-Angle formula can be used to convert such an angular measurement into a distance/size ratio.