Ingeometry, anangle is formed by twolines that meet at a point.[1] Each line is called aside of the angle, and the point they share is called thevertex of the angle.[2][3] The termangle is used to denote both geometric figures and theirsize or magnitude as associatedquantity.Angular measure ormeasure of angle are sometimes used to distinguish between the measure of the quantity and figure itself. The measurement of angles is intrinsically linked with circles and rotation, and this is often visualized or defined using thearc of acircle centered at the vertex and lying between the sides.
There is no universally agreed definition of an angle.[4] Angles can be conceived of and used in a variety of ways and while valid definitions may be given for specific contexts, it is difficult to give a single formal definition that is completely satisfactory in capturing all aspects of the general concept ofangle.[5]
One standard definition is that an angle is afigure consisting of two rays which lie in a plane and share a common endpoint. Alternatively, given such a figure, an angle might be defined as: theopening between the rays; thearea of the plane that lies between the rays; or theamount of rotation about the vertex of one ray to the other.
More generally, angles are also formed wherever twoline segments come together, such as at the corners of triangles and other polygons,[2] or at the intersection of two planes orcurves, in which case the rays lyingtangent to each curve at the point of intersection define the angle.[6]
It is common to consider that the sides of the angle divide the plane into two regions called theinterior of the angle and theexterior of the angle. The interior of the angle is also referred to as anangular sector.[7][8][a]
is formed by rays and. is the conventional measure of and is an alternative measure.
An angle symbol ( or, read as "angle") together with one or three defining points is used to identify angles in geometric figures. For example, the angle with vertex A formed by therays and is denoted as (using the vertex alone) or (with the vertex always named in the middle). The size or measure of the angle is denoted or.
In geometric figures andmathematical expressions, it is also common to useGreek letters (α,β,γ,θ,φ, ...) or lower case Roman letters (a, b, c, ...) asvariables to represent the size of an angle.[12] Angular measure is commonly ascalar quantity,[13] although in physics and some fields of mathematics,signed angles are used by convention to indicate a direction of rotation: positive for anti-clockwise; negative for clockwise.[14]
Angles are measured in various units, the most common being thedegree (denoted by the symbol°),radian (denoted by the symbolrad) andturn. These units differ in the way they divide up afull angle, an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its startingposition.[15]
Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360°.[16] A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see§ Measuring angles), but in such a way that its measure is 2π rad, approximately 6.28 rad.[17]
Historically the degree unit was chosen such as the straight angle or half the full angle was attributed the value of 180.
The angle addition postulate defines addition and subtraction of angles:θ +α =φ;φ −α =θ.
Theangle addition postulate states that if D is a point lying in the interior of then:[18] This relationshipdefines what it means to add any two angles: their vertices are placed together while sharing a side to create a new larger angle. The measure of the new larger angle is the sum of the measures of the two angles. Subtraction follows from rearrangement of the formula.[18]
"Vertical angle" redirects here; not to be confused withZenith angle.
Angles A and B are adjacent.
Angles A and B, and pair C and D are two pairs of vertical angles.Hatch marks indicate equality between pairs.
Adjacent angles (abbreviatedadj. ∠s), are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm".
Vertical angles are formed when two straight lines intersect at a point producing four angles. A pair of angles opposite each other are calledvertical angles,opposite angles orvertically opposite angles (abbreviatedvert. opp. ∠s),[22] where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. A theorem states that vertical angles are always congruent or equal to each other.[23]Atransversal is a line that intersects a pair of (often parallel) lines and is associated withexterior angles,interior angles,alternate exterior angles,alternate interior angles,corresponding angles, andconsecutive interior angles.[24]
When summing two angles (either adjacent or separated in space), three special cases are namedcomplementary,supplementary, andexplementary angles.
Complementary angles are angle pairs whose measures sum to a right angle (1/4 turn, 90°, orπ/2 rad). If the two complementary angles are adjacent, their non-shared sides form a right angle. In aright-angle triangle the two acute angles are complementary as the sum of the internal angles of atriangle is 180°. The difference between an angle and a right angle is termed thecomplement of the angle.[6]
Supplementary angles sum to a straight angle (1/2 turn, 180°, orπ rad). If the two supplementary angles areadjacent, their non-shared sides form a straight angle orstraight line and are called alinear pair of angles.[25] The difference between an angle and a straight angle is termed thesupplement of the angle.[26]
Explementary angles orconjugate angles sum to a full angle (1 turn, 360°, or 2π radians).[27] The difference between an angle and a full angle is termed theexplement orconjugate of the angle.[28][29]
Examples of non-adjacent supplementary angles include the consecutive angles of aparallelogram and opposite angles of acyclic quadrilateral. For a circle with center O, andtangent lines from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.
An angle that is part of asimple polygon is called aninterior angle if it lies on the inside of that simple polygon. A simpleconcave polygon has at least one interior angle, that is, a reflex angle. InEuclidean geometry, the measures of the interior angles of atriangle add up toπ radians, 180°, or1/2 turn; the measures of the interior angles of a simpleconvexquadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convexpolygon withn sides add up to (n − 2)π radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)1/2 turn.
The supplement of an interior angle is called anexterior angle; that is, an interior angle and an exterior angle form alinear 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.[30] If the corresponding interior angle is a reflex angle, the exterior angle should be considerednegative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick anorientation of theplane (orsurface) 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 asupplementary exterior angle. Exterior angles are commonly used inLogo Turtle programs when drawing regular polygons.
In atriangle, thebisectors of two exterior angles and the bisector of the other interior angle areconcurrent (meet at a single point).[31]: 149
In a triangle, three intersection points, each of an external angle bisector with the oppositeextended side, arecollinear.[31]: 149
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.[31]: 149
Some authors use the nameexterior angle of a simple polygon to mean theexplement exterior angle (not supplement!) of the interior angle.[32] This conflicts with the above usage.
The angle between twoplanes (such as two adjacent faces of apolyhedron) is called adihedral angle.[6] It may be defined as the acute angle between two linesnormal to the planes.
The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and thenormal to the plane.
Angle measurement encompasses both direct physical measurement using a measuring instrument such as aprotractor, as well as the theoretical calculation of angle size from other known quantities. While the measurement of angles is intrinsically linked with rotation and circles, there are various perspectives as to exactlywhat is being measured, including amongst others: the amount of rotation about the vertex of one ray to the other;[33] the amount of opening between the rays;[34] or the length of the arc that subtends the angle at the centre of a unit circle.[35]
The measurement of angles is inherently different from the measurement of other physical quantities such as length.[36] Angles of special significance (such as the right angle) inform the systems and units of angular measurement, which is not the case for length where the units of measurement (metres, feet) are arbitrary.
Broadly there are two approaches to measuring angles: relative to a reference angle (such as a right angle); and circular measurement.
A chosen reference angle (right angle, straight angle or full angle) can be divided into equal parts, and the size of one part used as a unit for measurement of other angles.
In the most common method of practical angular measurement a right angle is divided into 90 equal parts calleddegrees, while in the rarely usedcentesimal system, a right angle is divided into 100 equal parts calledgradians.[37][38]
The angle size can be measured ass/r radians ors/C turns
With circular measurement an angle is placed within a circle of any size, with the vertex at the circle's centre and the sides intersecting the perimeter.
Anarc with lengths is formed as the perimeter between the two points of intersection, which is said tosubtend the angle. The lengths can be used to measure the angle's sizeθ, however ass is dependent on the size of the circle chosen, the measure must be scaled. This can be done by taking the ratio ofs to either the radiusr or circumferenceC of the circle.
The ratio of the lengths by the radiusr is the number ofradians in the angle,[35] while the ratio of lengths by the circumferenceC is the number ofturns:
The radian measure ofθ
The value ofθ thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratioss/r ands/C are unaltered.[nb 1]
The ratios/r is called the "radian measure"[18] or "circular measure"[39][38][40] of an angle, but is also used to define aunit of measurement called a radian, which is defined as an angle for which the ratios/r = 1.[39] Thus, the measurement of an angle given bys/r can be thought of in two ways: firstly as a measure in terms of the angle's own proportions (ratio of arc length to radius), or secondly as the quantity of units in the angle (ratio of arc length of measured angle to arc length of unit angle).[41][38]
Thegrad, also calledgrade,gradian, orgon, is defined such that a right angle is equal to as 100 gradians. The grad is used mostly intriangulation and continentalsurveying.
Theminute of arc (orarcminute, or justminute) is a sexagesimal subunit of a degree. Often, latitude and longitude values are given in degrees, arcminutes, and arcseconds.
Thesecond of arc (orarcsecond, or justsecond) is a sexagesimal subunit of a minute of arc. Often, latitude and longitude values are given in degrees, arcminutes, and arcseconds.
The milliradian is a thousandth of a radian. For artillery and navigation a unit is used, often called a 'mil', which areapproximately equal to a milliradian. One turn is exactly 6000, 6300, or 6400 mils, depending on which definition is used.
In mathematics and theInternational System of Quantities, an angle is defined as a dimensionless quantity,[42] and in particular, theradian is defined as dimensionless in theInternational System of Units.[43] This convention prevents angles providing information fordimensional analysis. For example, when one measures an angle in radians by dividing the arc length by the radius, one is essentially dividing a length by another length, and the units of length cancel each other out. Therefore the result—the angle—doesn't have a physical "dimension" like meters or seconds.[44] This holds true with all angle units, such as radians, degrees, or turns—they all represent a pure number quantifying how much something has turned.[45] This is why, in many equations, angle units seem to "disappear" during calculations, which feels inconsistent and can lead to mixing up angle units.[46][47]
This has led to significant discussion among scientists and educators. Some scientists have suggested treating the angle as having its own fundamental dimension, similar to length or time.[48] This would mean that angle units like radians would always be explicitly present in calculations, facilitating dimensional analysis. However, this approach would also require changing many well-known mathematical and physics formulas, making them longer and perhaps a bit less familiar.[49] For now, the established practice is to write angle units where appropriate but consider them dimensionless, understanding that these units are important but behave differently from meters or kilograms.[50]
An angle denoted as∠BAC 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, It is therefore frequently helpful to impose a convention that allows positive and negative angular values to representorientations and/orrotations in opposite directions or "sense" relative to some reference.
In a two-dimensionalCartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. Theinitial side is on the positivex-axis, while the other side orterminal side is defined by the measure from the initial side in radians, degrees, or turns, withpositive angles representing rotations toward the positivey-axis andnegative angles representing rotations toward the negativey-axis. When Cartesian coordinates are represented bystandard position, defined by thex-axis rightward and they-axis upward, positive rotations areanticlockwise, and negative cycles areclockwise.
In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minusθ". For example, an orientation represented as −45° is effectively equal to an orientation defined as360° − 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 anorientation, which is typically determined by anormal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
Innavigation,bearings orazimuth 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°.
Angles that have the same measure (i.e., the same magnitude) are said to beequal orcongruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., allright angles are equal in measure).
Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are calledcoterminal angles.
Thereference angle (sometimes calledrelated angle) for any angleθ in standard position is the positive acute angle between the terminal side ofθ and the x-axis (positive or negative).[51] Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitudemodulo1/2 turn, 180°, orπ 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°).
For an angular unit, it is definitional that theangle addition postulate holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate:
Theslope orgradient is equal to thetangent of the angle and is often expressed as a percentage ("rise" over "run"). 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. Anelevation grade is a slope used to indicate the steepness of roads, paths and railway lines.
Thespread between two lines is defined inrational 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 oftrigonometric functions, such as thesine of the angle.
The angle between the two curves at P is defined as the angle between the tangentsA andB atP.
The angle between a line and acurve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between thetangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr.ἀμφί, on both sides, κυρτός, convex) orcissoidal (Gr. κισσός, ivy), biconvex;xystroidal orsistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex;amphicoelic (Gr. κοίλη, a hollow) orangulus lunularis, biconcave.[52]
Theancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only acompass and straightedge but could only trisect certain angles. In 1837,Pierre Wantzel showed that this construction could not be performed for most angles.
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from theirnormal vectors and betweenskew lines from their vector equations.
To define angles in an abstract realinner product space, we replace the Euclidean dot product (· ) by the inner product, i.e.
In a complexinner product space, the expression for the cosine above may give non-real values, so it is replaced withor, more commonly, using the absolute value, with
The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces and spanned by the vectors and correspondingly.
The definition of the angle between one-dimensional subspaces and given byin aHilbert space can be extended to subspaces of finite number of dimensions. Given two subspaces, with, this leads to a definition of angles called canonical orprincipal angles between subspaces.
InRiemannian geometry, themetric tensor is used to define the angle between twotangents. WhereU andV are tangent vectors andgij are the components of the metric tensorG,
The wordangle comes from theLatin wordangulus, meaning "corner".Cognate words include theGreekἀγκύλος (ankylοs) meaning "crooked, curved" and theEnglish word "ankle". Both are connected with theProto-Indo-European root*ank-, meaning "to bend" or "bow".[54]
Philosophers have debated the nature of angles for millennia, with some arguing that angles are a measure (quantity), and others saying they are a kind of shape defined by the lines that bound it (qualitative relation), and still others saying an angle is both.[55] Pedagogically, the accepted answer is that angles are defined as figures, and the measure of an angle is defined as the number of congruent non-overlapping copies of a unit angle necessary to cover the angle and its interior. Angles are said to be equal in measure andsimilar orcongruent in shape.[56]
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 metaphysicianProclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used byEudemus of Rhodes, who regarded an angle as a deviation from astraight line; the second, angle as quantity, byCarpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.[57]
The equality of vertically opposite angles is called thevertical angle theorem.Eudemus of Rhodes attributed the proof toThales of Miletus.[58][23] 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,[23] 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 angleA equalsx, the measure of angleC would be180° −x. Similarly, the measure of angleD would be180° −x. Both angleC and angleD have measures equal to180° −x and are congruent. Since angleB is supplementary to both anglesC andD, either of these angle measures may be used to determine the measure of AngleB. Using the measure of either angleC or angleD, we find the measure of angleB to be180° − (180° −x) =x. Therefore, both angleA and angleB have measures equal tox and are equal in measure.
Inastronomy, a given point on thecelestial sphere (that is, the apparent position of an astronomical object) can be identified using any of severalastronomical coordinate systems, where the references vary according to the particular system. Astronomers measure theangular separation of twostars by imagining two lines through the center of theEarth, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.
Astronomers also measure objects'apparent size as anangular diameter. For example, thefull moon has an angular diameter of approximately 0.5°, or 30 arcminutes, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." Thesmall-angle formula can convert such an angular measurement into a distance/size ratio.
Other astronomical approximations include:
0.5° is the approximate angular diameter of theSun and of theMoon as viewed from Earth.
1° is the approximate angular width of thelittle finger at arm's length.
10° is the approximate angular width of a closed fist at arm's length.
20° is the approximate angular width of a handspan at arm's length.
These measurements depend on the individual subject, and the above should be treated as roughrule of thumb approximations only.
In astronomy,right ascension is usually measured in angular units that are expressed in terms of time based on a 24-hour day.[59]
^This approach requires, however, an additional proof that the measure of the angle does not change with changing radiusr, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. SeeDimitrić (2012), for instance.
^An angular sector can be constructed by the combination of two rotatedhalf-planes, either their intersection or union (in the case of acute or obtuse angles, respectively).[9][10] It corresponds to acircular sector of infinite radius and a flatpencil of half-lines.[11]
^Heath, Thomas Little; Heiberg, J. L. (Johan Ludvig) (1908),The thirteen books of Euclid's Elements, Cambridge, The University Press, p. 176,A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
^Linton, John Alexander (1973),Phase and amplitude variation of Chandler wobble (Thesis), University of British Columbia,doi:10.14288/1.0052929,The latitude of a point on earth is defined as the conjugate of the angle between the point where the rotation axis pierces the celestial sphere (celestial pole) and the point where the local vertical pierces the same sphere (zenith).
^Quincey, Paul; Mohr, Peter J.; Phillips, William D. (2019), "Angles are inherently neither length ratios nor dimensionless",Metrologia,56 (4): 043001,arXiv:1909.08389,doi:10.1088/1681-7575/ab27d7
^Oberhofer, E. S. (1992), "What happens to the 'radians'?",The Physics Teacher,30 (3):170–171,doi:10.1119/1.2343500
^Quincey, Paul; Brown, Richard J C (2016), "Implications of adopting plane angle as a base quantity in the SI",Metrologia,53 (3):998–1002,arXiv:1604.02373,doi:10.1088/0026-1394/53/3/998
^Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (1993), "The radian—That troublesome unit",The Physics Teacher,31 (2):84–87,doi:10.1119/1.2343667
^McKeague, Charles P. (2008),Trigonometry (6th ed.), Belmont, CA: Thomson Brooks/Cole, p. 110,ISBN978-0495382607
^Morrow, Glenn Raymond (1992) [1970],A commentary on the first book of Euclid's Elements, pp. 98–104,the angle as such is none of the things mentioned but exists as a combination of all these categories
^Snell, Ronald Lee; Kurtz, Stanley E.; Marr, Jonathan M. (2019), "Introductory Material",Fundamentals of radio astronomy: astrophysics, Boca Raton, FL: CRC Press, Taylor & Francis Group,ISBN9781498725798
Dimitrić, Radoslav M. (2012),"On Angles and Angle Measurements"(PDF),The Teaching of Mathematics,XV (2):133–140,archived(PDF) from the original on 2019-01-17, retrieved2019-08-06
Henderson, David W.; Taimina, Daina (2005),Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104,ISBN978-0-13-143748-7
Heiberg, Johan Ludvig (1908),Heath, T. L. (ed.),Euclid, The Thirteen Books of Euclid's Elements, vol. 1,Cambridge: Cambridge University Press.
Shute, William G.; Shirk, William W.; Porter, George F. (1960),Plane and Solid Geometry, American Book Company, pp. 25–27
Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines",New Century Mathematics, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163,ISBN978-0-19-800177-5
Brinsmade, J. B. (1936), "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly",American Journal of Physics,4 (4):175–179,doi:10.1119/1.1999110
Leonard, B. P. (2021), "Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)",Metrologia,58 (5): 052001,doi:10.1088/1681-7575/abe0fc
Lévy-Leblond, Jean-Marc (1998), "Dimensional angles and universal constants",American Journal of Physics,66 (9):814–815,doi:10.1119/1.18964,ResearchGate:253371022
Mills, Ian (2016), "On the units radian and cycle for the quantity plane angle",Metrologia,53 (3):991–997,doi:10.1088/0026-1394/53/3/991
Quincey, Paul (2016), "The range of options for handling plane angle and solid angle within a system of units",Metrologia,53 (2):840–845,doi:10.1088/0026-1394/53/2/840
