The radian is the standard unit of angular measure, used in many areas of mathematics. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. 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. The SI unit of solid angle measurement is the steradian.
The radian 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 "1.2^{c}" (the second symbol is often mistaken for a degree: "1.2°"). As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, 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.
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Definition
One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length 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 = rθ.
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 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.
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