InEuclidean geometry, anangle orplane angle is the figure formed by tworays, called thesides of the angle, sharing a common endpoint, called thevertex of the angle.[1]Two intersectingcurves may also define an angle, which is the angle of the rays lyingtangent to the respective curves at their point of intersection. Angles are also formed by the intersection of two planes; these are calleddihedral angles. In any case, the resulting angle lies in aplane (spanned by the two rays or perpendicular to the line ofplane-plane intersection).
Themagnitude of an angle is called anangular measure or simply "angle". Two different angles may have the same measure, as in anisosceles triangle. "Angle" also denotes theangular sector, the infinite region of the plane bounded by the sides of an angle.[2][3][a]
Angle of rotation is ameasure conventionally defined as the ratio of acircular arc length to itsradius, and may be anegative number. In the case of an ordinary angle, the arc is centered at the vertex and delimited by the sides. In the case of an angle ofrotation, the arc is centered at the center of the rotation and delimited by any other point and itsimage after the rotation.
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".[7]
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.[8]
Inmathematical expressions, it is common to useGreek letters (α,β,γ,θ,φ, . . . ) asvariables denoting the size of some angle[9] (the symbolπ is typically not used for this purpose to avoid confusion with theconstant 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 therays AB and AC (that is, the half-lines from point A through points B and C) is denoted∠BAC or. 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,∠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, where the direction in which the angle is measured determines its sign (see§ Signed angles). 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,∠BAC always refers to the anticlockwise (positive) angle from B to C about A and∠CAB the anticlockwise (positive) angle from C to B about A.
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.
"Vertical angle" redirects here; not to be confused withZenith angle.
When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.
A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are calledvertical angles oropposite angles orvertically opposite angles. They are abbreviated asvert. opp. ∠s.[13]
The equality of vertically opposite angles is called thevertical angle theorem.Eudemus of Rhodes attributed the proof toThales of Miletus.[14][15] 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,[15] 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) = 180° − 180° +x =x. Therefore, both angleA and angleB have measures equal tox and are equal in measure.
AnglesA andB are adjacent.
Adjacent angles, often abbreviated asadj. ∠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". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively calledcomplementary,supplementary, andexplementary angles (see§ Combining angle pairs below).
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.[16]
Theangle addition postulate states that if B is in the interior of angle AOC, then
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:
Thecomplementary anglesa andb (b is thecomplement ofa, anda is the complement ofb.)
Complementary angles are angle pairs whose measures sum to one right angle (1/4 turn, 90°, orπ/2 radians).[17] If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of atriangle is 180 degrees, and the right angle accounts for 90 degrees.
The adjective complementary is from the Latincomplementum, associated with the verbcomplere, "to fill up". An acute angle is "filled up" by its complement to form a right angle.
The difference between an angle and a right angle is termed thecomplement of the angle.[18]
If anglesA andB are complementary, the following relationships hold:
(Thetangent of an angle equals thecotangent of its complement, and its secant equals thecosecant of its complement.)
Theprefix "co-" in the names of some trigonometric ratios refers to the word "complementary".
The anglesa andb aresupplementary angles.
Two angles that sum to a straight angle (1/2 turn, 180°, orπ radians) are calledsupplementary angles.[19]
If the two supplementary angles areadjacent (i.e., have a commonvertex and share just one side), their non-shared sides form astraight line. Such angles are called alinear pair of angles.[20] However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of aparallelogram are supplementary, and opposite angles of acyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.
If a point P is exterior to a circle with center O, and if thetangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.
Angles AOB and COD are conjugate as they form a complete angle. Considering magnitudes, 45° + 315° = 360°.
Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are calledexplementary angles orconjugate angles.[21]
The difference between an angle and a complete angle is termed theexplement of the angle orconjugate of an angle.
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.[22] 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).[23]: 149
In a triangle, three intersection points, each of an external angle bisector with the oppositeextended side, arecollinear.[23]: 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.[23]: 149
Some authors use the nameexterior angle of a simple polygon to mean theexplement exterior angle (not supplement!) of the interior angle.[24] This conflicts with the above usage.
The angle between twoplanes (such as two adjacent faces of apolyhedron) is called adihedral angle.[18] 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.
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to beequalcongruent orequal in measure.
In some contexts, such as identifying a point on a circle or describing theorientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a fullturn are effectively equivalent. In other contexts, such as identifying a point on aspiral curve or describing an object'scumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.
The measure of angleθ iss/r radians.
To measure an angleθ, acircular arc centered at the vertex of the angle is drawn, e.g., with a pair ofcompasses. The ratio of the lengths of the arc by the radiusr of the circle is the number ofradians in the angle:[25]Conventionally, in mathematics and theSI, the radian is treated as being equal to thedimensionless unit 1, thus being normally omitted.
The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the formk/2π, wherek is the measure of a complete turn expressed in the chosen unit (for example,k = 360° fordegrees or 400 grad forgradians):
The value ofθ 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 ratios/r is unaltered.[nb 1]
Throughout history, angles have beenmeasured in variousunits. These are known asangular units, with the most contemporary units being thedegree ( ° ), theradian (rad), and thegradian (grad), though many others have been used throughouthistory.[27] Most units of angular measurement are defined such that oneturn (i.e., the angle subtended by the circumference of a circle at its centre) is equal ton units, for some whole numbern. Two exceptions are the radian (and its decimal submultiples) and the diameter part.
Theradian is determined by the circumference of a circle that is equal in length to the radius of the circle (n = 2π = 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 israd. One turn is 2π radians, and one radian is180°/π, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unitrad 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 thetrigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in theSI.
Thedegree, denoted by a small superscript circle (°), is 1/360 of a turn, so oneturn is 360°. One advantage of this oldsexagesimal 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 forgeographical coordinates and inastronomy andballistics (n = 360)
Theminute of arc (orMOA,arcminute, or justminute) is1/60 of a degree =1/21,600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30/60 = 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 + 5.72/60 degrees. Anautical mile was historically defined as an arcminute along agreat circle of the Earth. (n = 21,600).
Thesecond of arc (orarcsecond, or justsecond) is1/60 of a minute of arc and1/3600 of a degree (n = 1,296,000). 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 arcsecond is the angle used to measure aparsec
Thegrad, also calledgrade,gradian, orgon. It is a decimal subunit of the quadrant. A right angle is 100 grads. Akilometre was historically defined as acenti-grad of arc along ameridian of the Earth, so the kilometer is the decimal analog to thesexagesimalnautical mile (n = 400). The grad is used mostly intriangulation and continentalsurveying.
The astronomicalhour angle is1/24 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 calledminute of time andsecond of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° =π/12 rad =1/6 quad =1/24 turn =16+2/3 grad.
Thepoint orwind, used innavigation, is1/32 of a turn. 1 point =1/8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided into four quarter points, so one turn equals 128.
The true milliradian is defined as a thousandth of a radian, which means that a rotation of oneturn would equal exactly 2000π mrad (or approximately 6283.185 mrad). Almost allscope sights forfirearms are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which areapproximately 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 as1/6400 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 (2π/6400 = 0.0009817... ≈1/1000).
Thebinary degree, also known as thebinary radian orbrad orbinary angular measurement (BAM).[28] The binary degree is used in computing so that an angle can be efficiently represented in a singlebyte (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 ofn.
Onequadrant is a1/4 turn and also known as aright angle. The quadrant is the unit inEuclid's Elements. In German, the symbol∟ has been used to denote a quadrant. 1 quad = 90° =π/2 rad =1/4 turn = 100 grad.
Thesextant was the unit used by theBabylonians,[31][32] The degree, minute of arc and second of arc aresexagesimal subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is theangle of theequilateral triangle or is1/6 turn. 1 Babylonian unit = 60° =π/3 rad ≈ 1.047197551 rad.
hexacontade
60
6°
Thehexacontade is a unit used byEratosthenes. It equals 6°, so a whole turn was divided into 60 hexacontades.
Thepechus was aBabylonian unit equal to about 2° or2+1/2°.
diameter part
≈376.991
≈0.95493°
Thediameter part (occasionally used in Islamic mathematics) is1/60 radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
zam
224
≈1.607°
In old Arabia, aturn was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that aturn is 224 zam.
Plane angle may be defined asθ =s/r, whereθ is the magnitude in radians of the subtended angle,s is circular arc length, andr is radius. One radian corresponds to the angle for whichs =r, hence1 radian = 1 m/m = 1.[33] However,rad is only to be used to express angles, not to express ratios of lengths in general.[34] A similar calculation usingthe area of a circular sectorθ = 2A/r2 gives 1 radian as 1 m2/m2 = 1.[35] The key fact is that the radian is adimensionless unit equal to1. In SI 2019, the SI radian is defined accordingly as1 rad = 1.[36] It is a long-established practice in mathematics and across all areas of science to make use ofrad = 1.[37][38]
Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations".[39] For example, an object hanging by a string from a pulley will rise or drop byy =rθ centimetres, wherer is the magnitude of the radius of the pulley in centimetres andθ is the magnitude of the angle through which the pulley turns in radians. When multiplyingr byθ, the unit radian does not appear in the product, nor does the unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for theangular velocity of a rolling wheel,ω =v/r, radians appear in the units ofω but not on the right hand side.[40] Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".[41] Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".[42]
At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as abase unit of measurement for abase quantity (and dimension) of "plane angle".[44][45][46] Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for thearea of a circle,πr2. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations".[47] A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.[46]
In particular, Quincey identifies Torrens' proposal to introduce a constantη equal to 1 inverse radian (1 rad−1) in a fashion similar to theintroduction of the constantε0.[47][b] With this change the formula for the angle subtended at the center of a circle,s =rθ, is modified to becomes =ηrθ, and theTaylor series for thesine of an angleθ becomes:[46][48]where is the angle in radians.The capitalized functionSin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed,[48] whilesin is the traditional function onpure numbers which assumes its argument is a dimensionless number in radians.[49] The capitalised symbol can be denoted if it is clear that the complete form is meant.[46][50]
Current SI can be considered relative to this framework as anatural unit system where the equationη = 1 is assumed to hold, or similarly,1 rad = 1. Thisradian convention allows the omission ofη in mathematical formulas.[51]
Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.[52] For example, theBoost units library defines angle units with aplane_angle dimension,[53] andMathematica's unit system similarly considers angles to have an angle dimension.[54][55]
It is 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 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 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).[56][57] 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. Some quantities related to angles where the angle addition postulate does not hold include:
Theslope orgradient is equal to thetangent 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.
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 atP 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.[58]
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 with
or, 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 by
in aHilbert space can be extended to subspaces of finite 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,
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° 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 diameter of theSun and of theMoon as viewed from Earth.
1° is the approximate width of thelittle 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 roughrule of thumb approximations only.
In astronomy,right ascension anddeclination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.
^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. See Radoslav M. Dimitrić, for instance.[26]
^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).[4][5] It corresponds to acircular sector of infinite radius and a flatpencil of half-lines.[6]
^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.
^Quincey 2016, p. 844: "Also, as alluded to inMohr & Phillips 2015, the radian can be defined in terms of the areaA of a sector (A =1/2θr2), in which case it has the units m2⋅m−2."
^Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit".The Physics Teacher.31 (2):84–87.Bibcode:1993PhTea..31...84A.doi:10.1119/1.2343667.
Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly".American Journal of Physics.4 (4):175–179.Bibcode:1936AmJPh...4..175B.doi:10.1119/1.1999110.
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.
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
