Some common angles, measured in radians. All the polygons are regular polygons.

The radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.2 rad" or "1.2c" (the second symbol can be mistaken for a degree: "1.2°").

However, the radian is the de facto unit of angular measurement for mathematicians, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ° is used.

The radian was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit. The SI unit of solid angle measurement is the steradian.

## Definition

An angle of 1 radian subtends an arc equal in length to the radius of the circle.

One radian is the angle subtended at the center of a circle by an arc of length equal to the radius of the circle.

More generally, the magnitude in radians of any angle subtended by two radii is equal to the ratio of the length of the enclosed arc to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = .

It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.

## History

The concept of a radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714.[1] He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure.

The term radian first appeared in print on June 5, 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. He used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, vacillated between rad, radial and radian. In 1874, Muir adopted radian after a consultation with James Thomson.[2][3][4]

## Conversions

### Conversion between radians and degrees

As explained above under "Definition", one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π. For example,

Conversely, to convert from degrees to radians, multiply by π/180. For example,

You can also convert radians to revolutions by dividing number of radians by 2π.

The table shows the conversion of some common angles.

 Degrees 0° 30° 45° 60° 90° 180° 270° 360° Radians 0

2π radians are equal to one complete revolution, which is 400g. So, to convert from radians to grads multiply by 200/π, and to convert from grads to radians multiply by π/200. For example,

## Reasons why radians are preferred in mathematics

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. One important reason is that results involving trigonometric functions are simple and "natural" when the function's argument is expressed in radians. For example, the use of radians leads to the simple identity

,

which is the basis of many other elegant identities in mathematics, including

.

The trigonometric functions also have simpler series expansions when radians are used; for example, the following Taylor series for sin x:

If x were expressed in degrees then the series would contain messy factors involving powers of π/180.

## Dimensional analysis

Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.

Another way to see the dimensionlessness of the radian is in the series representations of the trigonometric functions, such as the Taylor series for sin x mentioned earlier:
If x had units, then the sum would be meaningless: the linear term x cannot be added to (or have subtracted) the cubic term . Therefore, x must be dimensionless.

## Use in physics

The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically measured in radians per second (rad/s). One revolution per second is equal to 2π radians per second.

Similarly, angular acceleration is often measured in radians per second per second (rad/s2).

## SI multiples

SI prefixes have limited use with radians. The milliradian (0.001 rad, or 1 mrad) is used in gunnery and general targeting, because it corresponds to 1 m at a range of 1000 m (at such small angles, the curvature can be considered negligible). The divergence of laser beams is also usually measured in milliradians. Smaller units, like microradians (μrads) and nanoradians (nrads) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles. However, the larger prefixes have no apparent utility, mainly because to exceed 2π radians is to begin the same circle (or revolutionary cycle) again.

## References

1. ^ Biography of Roger Cotes, The MacTutor History of Mathematics
2. ^ Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147–148
3. ^ Nature, 1910, Vol. 83, pp. 156, 217, and 459–460
4. ^ Earliest Known Uses of Some of the Words of Mathematics

Radian is an Austrian experimental music group.

Their music touches on instrumental rock, post-rock, jazz and electronica, and is notable for imitating some of the more demanding musical structures of intelligent dance music.
Radian is a fictional mutant character in the Marvel Comics Universe. His first appearance was in New X-Men #135, created by Grant Morrison and Frank Quitely.

## Fictional character biography

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
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degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing 1360 of a full rotation.
De facto is a Latin expression that means "in fact" or "in practice" but not spelled out by law. It is commonly used in contrast to de jure (which means "by law") when referring to matters of law, governance, or technique (such as standards), that are found in the
degree symbol (°, Unicode: U+00B0, HTML: &deg;) is a typographical symbol, or glyph, that is used to represent degrees of arc (see Geographic coordinate system ) or temperature.

1°, 2°, 3°, etc.
Until 1995, SI (International System of Units) supplementary units were:

Quantity Symbol Name of SI supplementary unit Symbol for SI unit
19th century - 20th century - 21st century
1960s  1970s  1980s  - 1990s -  2000s  2010s  2020s
1992 1993 1994 - 1995 - 1996 1997 1998

Year 1995 (MCMXCV
SI derived units are part of the SI system of measurement units and are derived from the seven SI base units.

## Dimensionless derived units

The following SI units are actually dimensionless ratios, formed by dividing two identical SI units.
The solid angle, Ω, is the angle that an object subtends at a point. It is a measure of how big that object appears to an observer at that point. For instance, a small object nearby could subtend the same solid angle as a large object far away.
The steradian (symbol: sr) is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane.
angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
In geometry, an arc subtended by an angle is a curve whose endpoints are on the angle's two rays. The precise meaning varies with the context. For example, one may speak of the arc of a circle subtended by an angle whose vertex is a point on the circle.
circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, an arc is a segment of a circle. If the arc segment occupies a great circle (or great ellipse), it is considered a great-arc segment.
In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment. The radius is half the diameter.
Roger Cotes

Roger Cotes (1682-1716).
Born July 10, 1682
Burbage, Leicestershire
Died June 5, 1716
Cambridge, Cambridgeshire
Residence UK
Nationality British
17th century - 18th century - 19th century
1680s  1690s  1700s  - 1710s -  1720s  1730s  1740s
1711 1712 1713 - 1714 - 1715 1716 1717

:
Subjects:     Archaeology - Architecture -
5,5 Richter Scale, 34º36'00S, 57º53'59'W.
• 1900 - Second Boer War: British soldiers take Pretoria.
• 1907 - BAPS Swaminarayan religion established.
• 1915 - Denmark amends its constitution to allow women's suffrage.
• 18th century - 19th century - 20th century
1840s  1850s  1860s  - 1870s -  1880s  1890s  1900s
1870 1871 1872 - 1873 - 1874 1875 1876

:
Subjects:     Archaeology - Architecture -
James Thomson (February 16, 1822 - May 8, 1892) was an Irish engineer and physicist whose reputation would have been substantial had it not been overshadowed by that of his brother William Thomson, 1st Baron Kelvin.
William Thomson, 1st Baron Kelvin, OM, GCVO, PC, PRS, FRSE, (26 June 1824 – 17 December 1907) was a British mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century.
Queen's University Belfast (Irish: Ollscoil na Banríona, Béal Feirste) is a university in Belfast, Northern Ireland and a member of the Russell Group (a lobby group of major research universities in the United Kingdom).
Belfast
Irish - Béal Feirste

Pro Tanto Quid Retribuamus
"What shall we give in return for so much"

18th century - 19th century - 20th century
1840s  1850s  1860s  - 1870s -  1880s  1890s  1900s
1868 1869 1870 - 1871 - 1872 1873 1874

:
Subjects:     Archaeology - Architecture -
18th century - 19th century - 20th century
1830s  1840s  1850s  - 1860s -  1870s  1880s  1890s
1866 1867 1868 - 1869 - 1870 1871 1872

:
Subjects:     Archaeology - Architecture -