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 define also an angle, which is the angle of the tangents at the intersection point. For example, the spherical angle formed by two great circles on a sphere equals the dihedral angle between the planes containing the great circles.
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.
The word angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ? (ankyl?s), 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]}
In mathematical expressions, it is common to use Greek letters (?, ?, ?, ?, ?, . . . ) as variables denoting the size of some angle (to avoid confusion with its other meaning, the symbol ? is typically not used for this purpose). Lower case Roman letters (a, b, c, . . . ) are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples.
In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ?BAC (in Unicode ∠ ANGLE) 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.
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:
Name | zero | acute | right angle | obtuse | straight | reflex | 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} |
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 alternate interior angles, corresponding angles, interior angles, and exterior angles.^{[10]}
Three special angle pairs involve the summation of 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.
In order to measure an angle ?, 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/2?, where k is the measure of a complete turn expressed in the chosen unit (for example, for degrees or 400 grad for gradians):
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.^{[nb 1]}
In particular, the measure of angle is radian can be also interpreted as the arc length of its corresponding unit circle:^{[19]}
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.
Throughout history, angles have been measured in many different 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.^{[20]}
Angles expressed in radians are dimensionless for 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. The 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 quantity 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 2? 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 circle.
name | number in one turn | rotation angle | description |
---|---|---|---|
Turn | 1 | 360° | The turn, also cycle, full circle, revolution, and rotation, is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated τ, cyc, rev, or rot depending on the application.. The symbol τ can also be used as a mathematical constant to represent 2? radians. |
Multiples of ? | 2 | 180° | The multiples of ? (MUL?) unit is implemented in the RPN scientific calculator WP 43S.^{[21]}^{[22]}^{[23]} 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. It is the unit used in Euclid's Elements. 1 quad = 90° = ?/2 rad = 1/4 turn = 100 grad. |
Sextant | 6 | 60° | The sextant was the unit used by the Babylonians,^{[24]}^{[25]} 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° = ?/3 rad ? 1.047197551 rad. |
Radian | 2? | 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 180°/?, 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 | 6° | 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).^{[26]} 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 2^{n} equal parts for other values of n.
^{[27]} It is 1/256 of a turn.^{[26]} |
Degree | 360 | 1° | 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°/2?. |
Grad | 400 | 0°54? | The grad, also called grade, gradian, or gon. a right angle is 100 grads. It is a decimal subunit of the quadrant. 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. |
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 -? is effectively equivalent to an angle of "one full turn minus ?". 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°.
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.
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° | ?⁄12 | 1⁄24 | |
Minute | m | 0°15' | ?⁄720 | 1⁄1,440 | 1⁄60 hour |
Second | s | 0°0'15" | ?⁄43200 | 1⁄86,400 | 1⁄60 minute |
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:
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. ?, on both sides, , convex) or cissoidal (Gr. , ivy), biconvex; xystroidal or sistroidal (Gr. ?, a tool for scraping), concavo-convex; amphicoelic (Gr. , a hollow) or angulus lunularis, biconcave.^{[28]}
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.
In the Euclidean space, the angle ? 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.
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.
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.
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,
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.
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.
public domain: Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, 2 (11th ed.), Cambridge University Press, p. 14
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