Angle Explained

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius, and may be a negative number. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

History and etymology

The word angle comes from the Latin word, meaning "corner". Cognate words include the Greek Greek, Ancient (to 1453);: ἀγκύλος meaning "crooked, curved" and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".

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

Identifying angles

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) as variables denoting the size of some angle (the symbol is typically not used for this purpose to avoid confusion with the constant denoted by that symbol). Lower case Roman letters (a, b, c, . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples.

The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted or

\widehat{\rm BAC} . 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, 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 ). 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, always refers to the anticlockwise (positive) angle from B to C about A and the anticlockwise (positive) angle from C to B about A.

Types

Individual angles

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

• An angle equal to 0° or not turned is called a zero angle.
• An angle smaller than a right angle (less than 90°) is called an acute angle ("acute" meaning "sharp").
• An angle equal to  turn (90° or radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
• An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle ("obtuse" meaning "blunt").
• An angle equal to  turn (180° or radians) is called a straight angle.
• An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.
• An angle equal to 1 turn (360° or 2 radians) is called a full angle, complete angle, round angle or perigon.
• An angle that is not a multiple of a right angle is called an oblique angle.

The names, intervals, and measuring units are shown in the table below:

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

Vertical and angle pairs

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

Combining angle pairs

The angle addition postulate states that if B is in the interior of angle AOC, then

$$m\backslash angle\; \backslash mathrm\; =\; m\backslash angle\; \backslash mathrm\; +\; m\backslash angle\; \backslash mathrm$$

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:

Polygon-related angles

• An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle, that is, a reflex angle. In Euclidean geometry, the measures of the interior angles of a triangle add up to radians, 180°, or turn; the measures of the interior angles of a simple convex quadrilateral add up to 2 radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2) radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2) turn.
• The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure. In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
• In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).^[1]
• In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.^[1]
• 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.^[1]
• Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.^[2] This conflicts with the above usage.

Plane-related angles

• The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. 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.

Measuring angles

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 be equal congruent or equal in measure.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

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 ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle:$$\backslash theta\; =\; \backslash frac\; \backslash ,\; \backslash mathrm.$$Conventionally, in mathematics and the SI, the radian is treated as being equal to the dimensionless 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 form, where k is the measure of a complete turn expressed in the chosen unit (for example, for degrees or 400 grad for gradians):

$$\backslash theta\; =\; \backslash frac\; \backslash cdot\; \backslash frac.$$

The value of 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.

Units

Throughout history, angles have been measured in various units. These are known as angular units, with the most contemporary units being the degree (°), the radian (rad), and the gradian (grad), though many others have been used throughout history.^[3] Most units of angular measurement are defined such that one turn (i.e., the angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n. Two exceptions are the radian (and its decimal submultiples) and the diameter part.

In the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis.

The following table lists some units used to represent angles.

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

Signed angles

See main article: Angle of rotation.

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

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

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation, which is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

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

Equivalent angles

• Angles that have the same measure (i.e., the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all right angles are equal in measure).
• Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.
• The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).^{[4] [5]} Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo turn, 180°, or radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

Related quantities

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

• The slope or gradient is equal to the tangent of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.
• The spread between two lines is defined in rational geometry as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
• Although done rarely, one can report the direct results of trigonometric functions, such as the sine of the angle.

Angles between curves

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. Greek, Ancient (to 1453);: ἀμφί, 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.

Bisecting and trisecting angles

See main article: article and Angle trisection.

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

Dot product and generalisations

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

$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash cos(\backslash theta)\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; .$$

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

Inner product

To define angles in an abstract real inner product space, we replace the Euclidean dot product (·) by the inner product

\langle ⋅ , ⋅ \rangle , i.e.

$$\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; =\; \backslash cos(\backslash theta)\backslash \; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; .$$

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

$$\backslash operatorname\; \backslash left(\backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; \backslash right)\; =\; \backslash cos(\backslash theta)\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; .$$

or, more commonly, using the absolute value, with

$$\backslash left|\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; \backslash right|\; =\; \backslash left|\; \backslash cos(\backslash theta)\; \backslash right|\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; .$$

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces

\operatorname{span}( u ) and \operatorname{span}( v ) spanned by the vectors u and v correspondingly.

Angles between subspaces

The definition of the angle between one-dimensional subspaces

\operatorname{span}( u ) and \operatorname{span}( v ) given by

$$\backslash left|\; \backslash langle\; \backslash mathbf,\; \backslash mathbf\; \backslash rangle\; \backslash right|\; =\; \backslash left|\; \backslash cos(\backslash theta)\; \backslash right|\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |\; \backslash left\backslash |\; \backslash mathbf\; \backslash right\backslash |$$

in a Hilbert space can be extended to subspaces of finite dimensions. Given two subspaces

l{U} , l{W} with \dim ( l{U}) : = k \leq \dim ( l{W}) : = l , this leads to a definition of k angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry

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

$$\backslash cos\; \backslash theta\; =\; \backslash frac.$$

Hyperbolic angle

A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case.^[6] Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by Leonhard Euler in Introduction to the Analysis of the Infinite (1748).

Angles in geography and astronomy

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

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

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

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

Other astronomical approximations include:

• 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth.
• 1° is the approximate width of the little finger at arm's length.
• 10° is the approximate width of a closed fist at arm's length.
• 20° is the approximate width of a handspan at arm's length.

These measurements depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

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

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

See also

• Angle measuring instrument
• Angles between flats
• Angular statistics (mean, standard deviation)
• Angle bisector
• Angular acceleration
• Angular diameter
• Angular velocity
• Argument (complex analysis)
• Astrological aspect
• Central angle
• Clock angle problem
• Decimal degrees
• Dihedral angle
• Exterior angle theorem
• Golden angle
• Great circle distance
• Horn angle
• Inscribed angle
• Irrational angle
• Phase (waves)
• Protractor
• Solid angle
• Spherical angle
• Transcendent angle
• Trisection
• Zenith angle

Bibliography

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External links

• Angle . 2 . 29–30 . cs2. x .

Notes and References

1. Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007.
2. as cited in
3. Web site: angular unit. 2020-08-31. TheFreeDictionary.com.
4. Web site: Mathwords: Reference Angle. www.mathwords.com. 26 April 2018. live. https://web.archive.org/web/20171023035017/http://www.mathwords.com/r/reference_angle.htm. 23 October 2017.
5. Book: McKeague . Charles P. . Trigonometry . 2008 . Thomson Brooks/Cole . Belmont, CA . 978-0495382607 . 110 . 6th.
6. [Robert Baldwin Hayward]