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Learn more about "Angle"
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Angle
In geometry, an '''angle''' (in full, '''plane angle''') is the figure formed by two Ray (geometry)|rays sharing a common endpoint, called the vertex (geometry)|vertex of the angle.[Heath pp. 177-178 (for paragraph)]
Measuring angles
In order to measure an angle theta|θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of Compasses (drafting)|compasses. The length of the arc 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):
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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.
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.
Units
Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above. Of these units, treated in more detail below, the ''degree'' and the ''radian'' are by far the most common.
With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to ''n'' units, for some whole number ''n''. For example, in the case of degrees, A full circle of ''n'' units is obtained by setting in the formula above. (Proof. The formula above can be rewritten as One full circle, for which units, corresponds to an arc equal in length to the circle's circumference, which is 2π''r'', so . Substituting ''n'' for θ and 2π''r'' for ''s'' in the formula, results in )
- The '''degree (angle)|degree''', denoted by a small superscript circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for Geographic coordinate system|geographical coordinates and in astronomy and ballistics:
- The '''minute of arc''' (or '''MOA''', '''arcminute''', or just '''minute''') is 1/60 of a degree. 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 mile was historically defined as a minute of arc along a great circle 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.
- The '''radian''' is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle 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. 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 system.
- The '''angular mil|mil''' is ''approximately'' equal to a milliradian. There are several definitions.
- The '''full circle''' (or '''revolution''', '''rotation''', '''full Turn (geometry)|turn''' or '''cycle''') is one complete revolution. The revolution and rotation are abbreviated ''rev'' and ''rot'', respectively, but just ''r'' in ''Revolutions per minute|rpm'' (revolutions per minute). 1 full circle = 360° = 2''π'' rad = 400 gon = 4 right angles.
- The '''right angle''' is 1/4 of a full circle. It is the unit used in Euclid's Elements. 1 right angle = 90° = ''π''/2 rad = 100 gon.
- The '''angle of the equilateral triangle''' is 1/6 of a full circle. It was the unit used by the Babylonians, 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 '''grad (angle)|grad''', also called '''grade''', '''gradian''', or '''gon''' is 1/400 of a full circle, so one full circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right angle. A kilometre was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometre is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.
- The '''point''', used in navigation, is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is called "boxing the compass". 1 point = 1/8 of a right angle = 11.25° = 12.5 gon.
- The astronomical '''hour angle''' is 1/24 of a full circle. 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 right angle ≈ 16.667 gon.
- The '''binary degree''', also known as the '''Binary radians|Binary radian''' (or '''brad'''), is 1/256 of a full circle.[ooPIC Programmer's Guide ''www.oopic.com''] 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 allign to 2^n values for one whole turn.[Angles, integers, and modulo arithmetic Shawn Hargreaves ''blogs.msdn.com''] ''See also'' Binary scaling#Binary angle|Binary angle
- The '''grade (slope)|grade of a slope''', or '''gradient''', is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the tangent (trigonometric function)|tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians.
Positive and negative angles
A convention universally adopted in mathematical writing is that angles given a sign are '''positive angles''' if measured Clockwise and counterclockwise|anticlockwise, and '''negative angles''' if measured Clockwise and counterclockwise|clockwise, from a given line. If no line is specified, it can be assumed to be the 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 rotation 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 (geometric)|vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In navigation, bearing (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.
Approximations
- 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 Variable (mathematics)|variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol Pi|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|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
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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 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 '''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 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 '''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 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
- Astrological aspect
- Central angle
- Clock angle problem
- Complementary angles
- Great circle distance
- Hyperbolic angle
- Inscribed angle
- Protractor
- Solid angle for a concept of angle in three dimensions.
- Supplementary angles
References
- Euclid, commentary and trans. by T. L. Heath ''Elements'' Vol. 1 (1908 Cambridge) Google Books
External links
- Angle Bisectors in a Quadrilateral at cut-the-knot
- Constructing a triangle from its angle bisectors at cut-the-knot
- Convert angles in sexagesimal degree format to decimal degrees, and vice-versa
- Angle Estimation – for basic astronomy.
- 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)
Category:Elementary geometry
Category:Trigonometry
Category:Angle|
be-x-old:Кут
simple:Angle
zh-classical:角
Related Images- ∠, the angle symbol
- The angle θ is the quotient of s and r.
- θ = s/r rad = 1 rad.
Sources: StartLearningNow, Wikipedia | Usage license: GNU FDL
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