Angle

Overview

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an

**angle**is the figure formed by two rays sharing a common endpoint

Endpoint

An endpoint or end point is a mark of termination or completion.* In geometry, an endpoint is a point at which a line segment or ray terminates* Endpoint , the conclusion of a chemical reaction, particularly for titration...

, called the vertex

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

of the angle.

Angles are usually presumed to be in a Euclidean plane with the circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

of an circular arc

Arc (geometry)

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

. On another type of plane where a hyperbola

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

provides the standard of direction, the measure of an interval on the hyperbola is called hyperbolic angle

Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...

.

The word

*angle*comes from the Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

word

*angulus*, meaning "a corner".

Discussions

Encyclopedia

In geometry

, an

, called the vertex

of the angle.

Angles are usually presumed to be in a Euclidean plane with the circle

taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure

of an circular arc

. On another type of plane where a hyperbola

provides the standard of direction, the measure of an interval on the hyperbola is called hyperbolic angle

.

The word

word

words are the Greek

word "ankle

". Both are connected with the Proto-Indo-European

root

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, although his definitions of right, acute, and obtuse angles are certainly quantitative.

In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral

using polar coordinates, an extra full turn gives rise to a quite different point on the curve.

In order to measure an angle

The value of

, angles are considered to be dimensionless. There are several units used to measure angles. Of these units, listed below according to magnitude, the

Most units of angular measurement are defined such that one

, for which units, corresponds to an arc equal in length to the circle's circumference

, which is 2π

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

are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Negative bearings are not used in navigation, so north-west is 315 degrees.

The

. Gradients are often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

In rational geometry the

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

standing for 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 (a, b, c, ...) are also used. 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 or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").

Potentially, an angle denoted, 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, and 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 to the anticlockwise (positive) angle from C to B.

. In particular, if is a Euclidean angle, then using trigonometric functions we find,

and

for two numbers

To the ratio

If infinities are permitted for the quotient y/x one can define the angle θ as a function of x and y using the inverse tangent function for all points except the origin, assuming the inverse tangent varies from -π/2 to π/2,

The result will vary from -π to π. The values of x and y determine which quadrant the angle is in. Alternatively one can use the inverse cosine function assuming the result for the inverse cosine varies from 0 to π,

In this case the result will vary from 0 to 2π.

(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangent

s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—

, the angle θ between two vectors

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.

, we replace the Euclidean dot product (

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.

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.

, the metric tensor

is used to define the angle between two tangent

s. Where

, the location of any point on the Earth can be identified using a

and longitude

of any location in terms of angles subtended at the centre 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

s by imagining two lines through the centre 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.

Astronomers also measure the

. 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.

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an

**angle**is the figure formed by two rays sharing a common endpointEndpoint

An endpoint or end point is a mark of termination or completion.* In geometry, an endpoint is a point at which a line segment or ray terminates* Endpoint , the conclusion of a chemical reaction, particularly for titration...

, called the vertex

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

of the angle.

Angles are usually presumed to be in a Euclidean plane with the circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

of an circular arc

Arc (geometry)

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

. On another type of plane where a hyperbola

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

provides the standard of direction, the measure of an interval on the hyperbola is called hyperbolic angle

Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...

.

The word

*angle*comes from the LatinLatin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

word

*angulus*, meaning "a corner". The word*angulus*is a diminutive, of which the primitive form,*angus*, does not occur in Latin. CognateCognate

In linguistics, cognates are words that have a common etymological origin. This learned term derives from the Latin cognatus . Cognates within the same language are called doublets. Strictly speaking, loanwords from another language are usually not meant by the term, e.g...

words are the Greek

Greek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

*(ankylοs)*, meaning "crooked, curved," and the EnglishEnglish language

English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

word "ankle

Ankle

The ankle joint is formed where the foot and the leg meet. The ankle, or talocrural joint, is a synovial hinge joint that connects the distal ends of the tibia and fibula in the lower limb with the proximal end of the talus bone in the foot...

". Both are connected with the Proto-Indo-European

Proto-Indo-European language

The Proto-Indo-European language is the reconstructed common ancestor of the Indo-European languages, spoken by the Proto-Indo-Europeans...

root

**ank-*, meaning "to bend" or "bow".Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

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

Proclus

Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...

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

Carpus of Antioch

Carpus of Antioch was an ancient Greek mathematician. It is not certain when he lived; he may have lived any time between the 2nd century BC and the 2nd century AD. He wrote on mechanics, astronomy, and geometry. Proclus quotes from an Astronomical Treatise by Carpus concerning whether problems...

, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

## Measuring angles

Two angles are sometimes called congruent if there exists an isometry that transforms one of the angles into the other angle. The size of an angle is normally characterized by the smallest positive rotation that maps one of the rays into the other. Two angles are congruent if and only if they correspond to the same (smallest positive) rotation. Thus an angle as two rays is characterized by an angle of rotation. To avoid confusion when no isometry exists between particular representations of angles, angles that Euclid called "equal" are described as "equal in measure".In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral

Spiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

using polar coordinates, an extra full turn gives rise to a quite different point on the curve.

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 length of the arcTheta

Theta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth...

`s`is then divided by the radius of the circle`r`, and possibly multiplied by a scaling constant`k`(which depends on the units of measurement that are chosen):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.### Units

In dimensional analysisDimensional analysis

In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

, angles are considered to be dimensionless. There are several units used to measure angles. Of these units, listed below according to magnitude, the

*degree*

and theDegree (angle)

A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

*radian*

are by far the most common.Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

Most units of angular measurement are defined such that one

*turn*

(i.e. one full circle) is equal toTurn (geometry)

A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

*n*units, for some whole number*n*. The two exceptions are the radian and the diameter part. For example, in the case of degrees, A*turn*

ofTurn (geometry)

A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

*n*units is obtained by setting in the formula above. (Proof. The formula above can be rewritten as One turnTurn (geometry)

A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

, for which units, corresponds to an arc equal in length to the circle's circumference

Circumference

The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

, which is 2π

*r*, so . Substituting*n*for θ and 2π*r*for*s*in the formula, results in )- The
**turn**(orTurn (geometry)**full circle**,**revolution**,**rotation**, or**cycle**) is one full circle. ATurn (geometry)*rev*or*rot*depending on the application, but just*r*in*rpm*(revolutions per minute). 1Revolutions per minuteRevolutions per minute is a measure of the frequency of a rotation. It annotates the number of full rotations completed in one minute around a fixed axis...Turn (geometry)*π*rad = 400 grad = 4 right angles. - The
*quadrant*is 1/4 of a turnTurn (geometry)

, i.e. a**right angle**. It is the unit used in Euclid's ElementsRight angleIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...Euclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

. 1 quad. = 90° =*π*/2 rad = 1/4 turn = 100 grad. In German the symbol^{∟}has been used to denote a quadrant. - The
**angle of the equilateral triangle**is 1/6 of a turnTurn (geometry)

. It was the unit used by the BabyloniaBabyloniaBabylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...

ns, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° =*π*/3 rad ≈ 1.047197551 rad.

- The
**radian**is the angle subtended by an arc of a circle that has the same length as the circle's radius (RadianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...`k`= 1 in the formula given earlier). One turnTurn (geometry)

is 2*π*radians, and one radian is 180/*π*degrees, or about 57.2958 degrees. The radian is abbreviated*rad*, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. 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 functionTrigonometric functionIn mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SISiSi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

system. - The astronomical
**hour angle**is 1/24 of a turnHour angleIn astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the position of a point on the celestial sphere....Turn (geometry)

. Since 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**. Note that these are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° =*π*/12 rad = 1/6 quad. = 1/24Turn (geometry) - The
**point**, used in navigationCompass PointCompass Point may refer to:* Compass point, a direction on a traditional compass* Compass Point * Compass Point Shopping Centre, a shopping mall in Singapore* Compass Point Studios, a studio in Nassau, Bahamas...NavigationNavigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

, is 1/32 of a turnTurn (geometry)

. 1 point = 1/8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided in four quarter-points so that 1 turnTurn (geometry)

equals 128 quarter-points. - EratosthenesEratosthenesEratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...

used a unit of 6° so that a whole turnTurn (geometry)

was divided in 60 units. - The BabyloniansBabylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

sometimes used the unit pechus of about 2° or 2½°. - The
**binary degree**, also known as the**binary radian**(or**brad**), is 1/256 of a turnTurn (geometry)

. The binary degree is used in computing so that an angle can be efficiently represented in a single byteByteThe byte is a unit of digital information in computing and telecommunications that most commonly consists of eight bits. Historically, a byte was the number of bits used to encode a single character of text in a computer and for this reason it is the basic addressable element in many computer...

(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*. - The
**degree**, denoted by a small superscript circle (°), is 1/360 of a turnDegree (angle)A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...Turn (geometry)

, so oneTurn (geometry)Geographic coordinate systemA geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

and in astronomyAstronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and ballisticsBallisticsBallistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...

: - The
**diameter part**(occasionally used in Islamic mathematics) is 1/60 radian. One "diameter part" is approximately 0.95493 degrees. - The
**grad**, also calledGrad (angle)The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...**grade**,**gradian**, or**gon**, is 1/400 of a turn, so a right angle is 100 grads. It is a decimal subunit of the quadrant. A kilometreKilometreThe kilometre is a unit of length in the metric system, equal to one thousand metres and is therefore exactly equal to the distance travelled by light in free space in of a second...

was historically defined as a centiCentiCenti is a prefix in the metric system denoting a factor of one hundredth. Adopted in 1795, the prefix comes from the Latin centum, meaning hundred. It is mainly used with the metre to express lengths in centimetres, a common unit of length....

-grad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The grad is used mostly in triangulationTriangulationIn trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...

. - The
**mil**isAngular milAn angular mil, also mil, is a unit of angle. All versions of the angular mil are approximately the same size as a trigonometric milliradian.-History:The milliradian was first identified in the mid nineteenth Century...*approximately*equal to a milliradian. There are several definitions ranging from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes), with the milliradian being approximately 0.05729578 degrees (3.43775 minutes). - The
**minute of arc**(orMinute of arcA minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....**MOA**,**arcminute**, or just**minute**) is 1/60 of a degree = 1/3600 turnTurn (geometry)

. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 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 mileNautical mileThe nautical mile is a unit of length that is about one minute of arc of latitude along any meridian, but is approximately one minute of arc of longitude only at the equator...

was historically defined as a minute of arc along a great circleGreat circleA great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

of the Earth. - The
**second of arc**(or**arcsecond**, or just**second**) is 1/60 of a minute of arc 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.

### Positive and negative angles

In mathematics, the angle from the first to the second coordinate axis of a coordinate system is considered as positive. Therefore angles given a sign are**positive angles**if measured anticlockwise, and**negative angles**if measured clockwise, from a given line. If no line is specified, it can be assumed to be the first coordinate axis (x-axis) in the Cartesian plane. In many geometrical situations a negative angle of −*θ*is effectively equivalent to a positive angle of "one full turn less*θ*". For example, a clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to an anticlockwise rotation of 360° − 45° (that is, an angle of 315°).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

Navigation

Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

, bearings

Bearing (navigation)

In marine navigation, a bearing is the direction one object is from another object, usually, the direction of an object from one's own vessel. In aircraft navigation, a bearing is the actual compass direction of the forward course of the aircraft...

are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Negative bearings are not used in navigation, so north-west is 315 degrees.

### Alternative ways of measuring the size of an angle

There are several alternatives to measuring the size of an angle by the corresponding angle of rotation.The

**grade of a slope**, or**gradient**is equal to the tangent of the angle, or sometimes the sineSine

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

. Gradients are often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

In rational geometry the

*spread*between two lines is defined at the square of sine of the angle between the lines. Since 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 of the spread between the lines.### Astronomical approximations

Astronomers measure angular separation of objects in degrees from their point of observation.- 1° is approximately the width of a little finger at arm's length.
- 10° is approximately the width of a closed fist at arm's length.
- 20° is approximately the width of a handspan at arm's length.

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

## Identifying angles

In mathematical expressions, it is common to use Greek letters (`α`,`β`,`γ`,`θ`,`φ`, ...) to serve as variablesVariable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π

Pi

' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

is typically not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. 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 or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").

Potentially, an angle denoted, 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, and 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 to the anticlockwise (positive) angle from C to B.

## Types of angles

- An angle equal to 1/4 turn (90° or
*π*/2 radians) is called aPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...**right angle**.Right angleIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...- Two lines that form a right angle are said to be
**perpendicular**orPerpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...**orthogonal**.OrthogonalityOrthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

- Two lines that form a right angle are said to be
- Angles equal to 1/2 turn (180° or two right angles) are called
**straight angles**. - Angles equal to 1 turn (360° or four right angles) are called
**full angles**. - Angles that are not right angles or a multiple of a right angle are called
**oblique angles**. - Angles smaller than a right angle (less than 90°) are called
**acute angles**("acute" meaning "sharp"). - Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called
**obtuse angles**("obtuse" meaning "blunt"). - Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called
**reflex angles**. - Angles that have the same measure (i.e. the same magnitude) are said to be
**congruent**. Following this definition for congruent angles, an angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. allCongruence (geometry)In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...*right angles*are*congruent*). - Two angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called
**vertical angles**orVertical (angles)In geometry, a pair of angles is said to be vertical if the angles are formed from two intersecting lines and the angles are not adjacent. The two angles share a vertex...**opposite angles**or**vertically opposite angles**. These angles are equal in measure. - Angles that share a common vertex and edge but do not share any interior points are called
**adjacent angles**.Adjacent anglesIn geometry, adjacent angles, often shortened as adj. ∠s, are angles that have a common ray coming out of the vertex going between two other rays. In other words, they are angles that are side by side, or adjacent.- Complementary adjacent angles :... - Two angles that sum to one right angle (90°) are called
**complementary angles**.- The difference between an angle and a right angle is termed the
**complement**of the angle.

- The difference between an angle and a right angle is termed the
- Two angles that sum to a straight angle (180°) are called
**supplementary angles**.- The difference between an angle and a straight angle (180°) is termed the
**supplement**of the angle.

- The difference between an angle and a straight angle (180°) is termed the
- Two angles that sum to one turn (360°) are called
**explementary angles**or**conjugate angles**. - An angle that is part of a simple polygonSimple polygonIn geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....

is called an**interior angle**if it lies on the inside of that simple polygon. A concave simple polygon has at least one interior angle that exceeds 180°.- In Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, the measures of the interior angles of a triangleTriangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

add up to*π*radians, or 180°, or 1/2 turn; the measures of the interior angles of a simple quadrilateralQuadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

add up to 2*π*radians, or 360°, or 1 turn. In general, the measures of the interior angles of a simple polygonPolygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

with*n*sides add up to [(*n*− 2) ×*π*] radians, or [(*n*− 2) × 180]°, or (*2n*− 4) right angles, or (*n/2*− 1) turn.

- In Euclidean geometry
- The angle supplementary to the interior angle is called the
**exterior angle**. It measures the amount of rotation one has to make at this vertex to trace out 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, but one will have to pick an orientationOrientation (mathematics)In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

of the planePlane (mathematics)In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

(or surfaceSurfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

) to decide the sign of the exterior angle measure.- In Euclidean geometry, the sum of the exterior angles of a simple polygon will be one full turn (360°).

- Some authors use the name
**exterior angle**of a simple polygon to simply mean the explementary (*not*supplementary!) of the interior angle. This conflicts with the above usage. - The angle between two planesPlane (mathematics)In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

(such as two adjacent faces of a polyhedronPolyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

) is called a**dihedral angle**. It may be defined as the acute angle between two lines normalDihedral angleIn geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...Surface normalA surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

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.
- Alternate angles, corresponding angle, interior angles and exterior angles are associated with a transversalTransversal (geometry)In geometry, a transversal is a line that passes through two or more other lines in the same plane at different points. When the lines are parallel, as is often the case, a transversal produces several congruent and several supplementary angles...

of a pair of lines by a third. - A
**reference angle**is the acute version of any angle determined by repeatedly subtracting or adding 180 degrees, and subtracting the result from 180 degrees if necessary, until a value between 0 degrees and 90 degrees is obtained. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180-150). An angle of 750 degrees has a reference angle of 30 degrees (750-720).

### Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangleRight triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

. In particular, if is a Euclidean angle, then using trigonometric functions we find,

and

for two numbers

*x*and*y*. So an angle in the Euclidean plane can be legitimately given by two numbers*x*and*y*.To the ratio

*y*/*x*there correspond two angles in the geometric range 0 < θ < 2π, sinceIf infinities are permitted for the quotient y/x one can define the angle θ as a function of x and y using the inverse tangent function for all points except the origin, assuming the inverse tangent varies from -π/2 to π/2,

The result will vary from -π to π. The values of x and y determine which quadrant the angle is in. Alternatively one can use the inverse cosine function assuming the result for the inverse cosine varies from 0 to π,

In this case the result will vary from 0 to 2π.

## Angles between curves

The angle between a line and a curveCurve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—

*amphicyrtic*(Gr. , 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.## Dot product and generalisation

In the Euclidean planeEuclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, the angle θ between two vectors

**u**and**v**is related to their dot productDot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

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

Skew lines

In solid geometry, skew lines are two lines that do not intersect and are not parallel. Equivalently, they are lines that are not coplanar. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron...

from their vector equations.

## Inner product

To define angles in an abstract real inner product spaceInner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

, we replace the Euclidean dot product (

**·**) by the inner product , i.e.In a complex inner product space

Inner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

, 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 byin a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

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

Principal angles

In linear algebra , the principal angles, also called canonical angles, provide information about the relative position of two subspaces of a Euclidean space...

between subspaces.

## Angles in Riemannian geometry

In Riemannian geometryRiemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

is used to define the angle between two tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

s. Where

*U*and*V*are tangent vectors and*g*_{ij}are the components of the metric tensor*G*,## Angles in geography and astronomy

In geographyGeography

Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...

, the location of any point on the Earth can be identified using a

**geographic coordinate system**

. This system specifies the latitudeGeographic coordinate system

A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

Latitude

In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

and longitude

Longitude

Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

of any location in terms of angles subtended at the centre of the Earth, using the equator

Equator

An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

and (usually) the Greenwich meridian as references.

In astronomy

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, a given point on the celestial sphere

Celestial sphere

In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...

(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 theAstronomical coordinate systems

Astronomical coordinate systems are coordinate systems used in astronomy to describe the location of objects in the sky and in the universe. The most commonly occurring such systems are coordinate systems on the celestial sphere, but extragalactic coordinates systems are also important for...

**angular separation**of two starStar

A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s by imagining two lines through the centre of the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.

Astronomers also measure the

**apparent size**of objects as an angular diameterAngular diameter

The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its...

. For example, the full moon

Full moon

Full moon lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun. More precisely, a full moon occurs when the geocentric apparent longitudes of the Sun and Moon differ by 180 degrees; the Moon is then in opposition with the Sun.Lunar eclipses can only occur at...

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

- Angle bisector
- Argument (complex analysis)
- Astrological aspectAstrological aspectIn astrology, an aspect is an angle the planets make to each other in the horoscope, and also to the ascendant, midheaven, descendant, lower midheaven, and other points of astrological interest. Aspects are measured by the angular distance in degrees and minutes of ecliptic longitude between two...
- Central angleCentral angleA central angle is an angle which vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is equal to the central angle itself...
- Clock angle problemClock angle problemClock angle problems are a type of mathematical problem which involve finding the angles between the hands of an analog clock. Questions of this nature may appear in text books, tests, examinations or mathematics competitions.-Math problem:...
- Complementary anglesComplementary anglesIn geometry, complementary angles are angles whose measures sum to 90°. If the two complementary angles are adjacent their non-shared sides form a right angle....
- Great circle distance
- Inscribed angleInscribed angleIn geometry, an inscribed angle is formed when two secant lines of a circle intersect on the circle....
- ProtractorProtractorIn geometry, a protractor is a circular or semicircular tool for measuring an angle or a circle. The units of measurement utilized are usually degrees.Some protractors are simple half-discs; these have existed since ancient times...
- Solid angleSolid angleThe solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

for a concept of angle in three dimensions. - Supplementary anglesSupplementary anglesSupplementary angles are pairs of angles that add up to 180 degrees. Thus the supplement of an angle of x degrees is an angle of degrees....
- Irrational angle
- Angular velocityAngular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

## External links

- Angle Bisectors in a Quadrilateral at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Constructing a triangle from its angle bisectors at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Angle Estimation – for basic astronomyAstronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

. - Angle definition pages with interactive applets.
- Various angle constructions with compass and straightedge
- GonioLab DD – Convert between DecDeg and DegMinSec and vice-versa (requires Java Web Start)