It has been suggested that Angle of rotation be merged into this article. (Discuss) Proposed since March 2022.
An angle formed by two rays emanating from a vertex.
An angle formed by two rays emanating from a vertex.

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.[1] Angles formed by two rays 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.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. 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 angulus, meaning "corner"; cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".[2]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus, an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept.[3]

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 (to avoid confusion with its other meaning, the symbol π is typically not used for this purpose). Lower case Roman letters (abc, . . . ) 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.

In geometric figures, angles may also be identified by the three points that define them. For example, the angle with vertex A formed by the rays AB and AC (that is, the lines from point A to points B and C) is denoted ∠BAC or . Where there is no risk of confusion, the angle may sometimes be referred to simply by its vertex (in this case "angle A").

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

Types of angles

"Oblique angle" redirects here. For the cinematographic technique, see Dutch angle.

Individual angles

There is some common terminology for angles, whose measure is always non-negative (see § Positive and negative angles):[4][5]

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

Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Reflex angle
Name   zero acute right angle obtuse straight reflex perigon
Unit Interval
turn   0 turn (0, 1/4) turn 1/4 turn (1/4, 1/2) turn 1/2 turn (1/2, 1) turn 1 turn
radian 0 rad (0, 1/2π) rad 1/2π rad (1/2π, π) rad π rad (π, 2π) rad 2π rad
degree   (0, 90)° 90° (90, 180)° 180° (180, 360)° 360°
gon   0g (0, 100)g 100g (100, 200)g 200g (200, 400)g 400g

Equivalence angle pairs

Vertical and adjacent angle pairs

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.
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 with Zenith 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.

The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.[8][9] 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,[9] 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 angle A equals x, then the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.
Angles A and B are adjacent.
Angles A and B are adjacent.

A transversal is a line that intersects a pair of (often parallel) lines, and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.[10]

Combining angle pairs

Three special angle pairs involve the summation of angles:

The complementary angles a and b (b is the complement of a, and a is the complement of b).
The complementary angles a and b (b is the complement of a, and a is the complement of b).
The adjective complementary is from Latin complementum, associated with the verb complere, "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 the complement of the angle.[12]
If angles A and B are complementary, the following relationships hold:
(The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement.)
The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".
The angles a and b are supplementary angles.
The angles a and b are supplementary angles.
If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles.[14] However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic 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 the tangent 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 internal angles of a triangle is a straight angle.

Sum of two explementary angles is a complete angle.
Sum of two explementary angles is a complete angle.

Polygon-related angles

Internal and external angles.
Internal and external angles.

Plane-related angles

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 that have the same size are said to be equal or 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 the cumulative rotation of an object 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 angle θ is s/r radians.
The measure of angle θ is s/r radians.

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle. Conventionally, in mathematics and in the SI, the radian is treated as being equal to the dimensionless value 1.

The angle expressed another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form k/2π, where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians):

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.[nb 1]

Angle addition postulate

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

The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

Units

Definition of 1 radian
Definition of 1 radian

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.[19]

In the International System of Quantities, angle is defined as a dimensionless quantity. This impacts how angle is treated in dimensional analysis.

Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to n units, for some whole number n. Two exceptions are the radian (and its decimal submultiples) and the diameter part.

One radian is the angle subtended by an arc of a circle that has the same length as the circle's radius. The radian is the derived unit of angular measurement in the SI system. By definition, it is dimensionless, though it may be specified as rad to avoid ambiguity. Angles measured in degrees, are shown with the symbol °. Subdivisions of the degree are minute (symbol ′, 1′ = 1/60°) and second (symbol ″, 1″ = 1/3600°). An angle of 360° corresponds to the angle subtended by a full circle, and is equal to 2π radians, or 400 gradians.

Other units used to represent angles are listed in the following table. These units are defined such that the number of turns is equivalent to a full rotation.

name number in one turn in degrees description
Turn 1 360° The turn, also cycle, revolution, and rotation, is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated cyc, rev, or rot depending on the application. A turn is equal to 2π radians or 360 degrees.
Multiples of π 2 180° The multiples of π radians (MULπ) unit is implemented in the RPN scientific calculator WP 43S.[20][21][22] See also: IEEE 754 recommended operations
Quadrant 4 90° One quadrant is a 1/4 turn and also known as a right angle. The quadrant is the unit used in Euclid's Elements. In German, the symbol has been used to denote a quadrant. 1 quad = 90° = π/2 rad = 1/4 turn = 100 grad.
Sextant 6 60° The sextant was the unit used by the Babylonians,[23][24] The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is especially easy to construct with ruler and compasses. It is the angle of the equilateral triangle or is 1/6 turn. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.
Radian 2π 57°17′ The radian 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 is rad. One turn is 2π radians, and one radian is 180°/π, or about 57.2958 degrees. In mathematical texts, angles are often treated as being dimensionless with the radian equal to one, resulting in the unit rad often 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, which also treats angle as being dimensionless.
Hexacontade 60 The hexacontade is a unit used by Eratosthenes. It is equal to 6°, so that a whole turn was divided into 60 hexacontades.
Binary degree 256 1°33'45" The binary degree, also known as the binary radian or brad or binary angular measurement (BAM).[25] 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 angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n.

[26] It is 1/256 of a turn.[25]

Degree 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 are also in use, especially for geographical coordinates and in astronomy and ballistics (n = 360) The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. The case of degrees for the formula given earlier, a degree of n = 360° units is obtained by setting k = 360°/2π.
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.
Minute of arc 21,600 0°1′ The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. A nautical mile was historically defined as a minute of arc along a great circle of the Earth (n = 21,600). The arcminute is 1/60 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 also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as an arcminute along a great circle of the Earth.
Second of arc 1,296,000 0°0′1″ The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree (n = 1,296,000). The arcsecond (or second of arc, or just second) is 1/60 of an arcminute and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

Other descriptors

Signed angles

See also: Sign (mathematics) § Angles

Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions 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 rotations 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 equivalent to an orientation represented 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 relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings 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°.

Alternative ways of measuring the size of an angle

There are several alternatives to measuring the size of an angle by the angle of rotation. The slope or gradient is equal to the tangent of the angle, or sometimes (rarely) the sine; a gradient is often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of the angle in radians.

In rational geometry the spread between two lines is defined 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.

Astronomical approximations

Main article: Angular diameter

Astronomers measure angular separation of objects in degrees from their point of observation.

These measurements clearly 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 Symbol Degree Radians Circle Other
Hour h 15° π12 124
Minute m 0°15′ π720 11,440 160 hour
Second s 0°0′15″ π43200 186,400 160 minute

Measurements that are not angular units

Not all angle measurements are angular units, for an angular measurement, it is definitional that the angle addition postulate holds.

Some angle measurements where the angle addition postulate does not hold include:

Angles between curves

The angle between the two curves at P is defined as the angle between the tangents A and B at P.
The angle between the two curves at P is defined as the angle between the tangents A and B at P.

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. 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.[27]

Bisecting and trisecting angles

Main articles: Bisection § Angle bisector, 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 for most angles this construction cannot be performed.

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

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 , i.e.

In a complex inner 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 and thus describes the angle between one-dimensional subspaces and spanned by the vectors and correspondingly.

Angles between subspaces

The definition of the angle between one-dimensional subspaces and given by

in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces , with , this leads to a definition of 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 gij are the components of the metric tensor G,

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. 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 weaving of the two types of angle and function was explained by Leonhard Euler in Introduction to the Analysis of the Infinite.

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 can be measured and is the angular separation between the two stars.

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 the apparent size of objects 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 be used to convert such an angular measurement into a distance/size ratio.

See also

Notes

  1. ^ This approach requires however an additional proof that the measure of the angle does not change with changing radius r, 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.[18]

References

  1. ^ Sidorov 2001
  2. ^ Slocum 2007
  3. ^ Chisholm 1911; Heiberg 1908, pp. 177–178
  4. ^ "Angles – Acute, Obtuse, Straight and Right". www.mathsisfun.com. Retrieved 2020-08-17.
  5. ^ Weisstein, Eric W. "Angle". mathworld.wolfram.com. Retrieved 2020-08-17.
  6. ^ "Mathwords: Reference Angle". www.mathwords.com. Archived from the original on 23 October 2017. Retrieved 26 April 2018.
  7. ^ Wong & Wong 2009, pp. 161–163
  8. ^ Euclid. The Elements. Proposition I:13.
  9. ^ a b Shute, Shirk & Porter 1960, pp. 25–27.
  10. ^ Jacobs 1974, p. 255.
  11. ^ "Complementary Angles". www.mathsisfun.com. Retrieved 2020-08-17.
  12. ^ a b Chisholm 1911
  13. ^ "Supplementary Angles". www.mathsisfun.com. Retrieved 2020-08-17.
  14. ^ Jacobs 1974, p. 97.
  15. ^ Henderson & Taimina 2005, p. 104.
  16. ^ a b c Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007.
  17. ^ D. Zwillinger, ed. (1995), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, p. 270 as cited in Weisstein, Eric W. "Exterior Angle". MathWorld.
  18. ^ 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. Retrieved 2019-08-06.
  19. ^ "angular unit". TheFreeDictionary.com. Retrieved 2020-08-31.
  20. ^ Bonin, Walter (2016-01-11). "RE: WP-32S in 2016?". HP Museum. Archived from the original on 2019-08-06. Retrieved 2019-08-05.
  21. ^ Bonin, Walter (2019) [2015]. WP 43S Owner's Manual (PDF). 0.12 (draft ed.). pp. 72, 118–119, 311. ISBN 978-1-72950098-9. Retrieved 2019-08-05.[permanent dead link] [1] [2] (314 pages)
  22. ^ Bonin, Walter (2019) [2015]. WP 43S Reference Manual (PDF). 0.12 (draft ed.). pp. iii, 54, 97, 128, 144, 193, 195. ISBN 978-1-72950106-1. Retrieved 2019-08-05.[permanent dead link] [3] [4] (271 pages)
  23. ^ Jeans, James Hopwood (1947). The Growth of Physical Science. CUP Archive. p. 7.
  24. ^ Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2.
  25. ^ a b "ooPIC Programmer's Guide - Chapter 15: URCP". ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 2008-06-28. Retrieved 2019-08-05.
  26. ^ Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com. Archived from the original on 2019-06-30. Retrieved 2019-08-05.
  27. ^ Chisholm 1911; Heiberg 1908, p. 178

Bibliography

 This article incorporates text from a publication now in the public domainChisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, vol. 2 (11th ed.), Cambridge University Press, p. 14