Radian

Radian, in mathematics, central angle inscribed in a circle and subtended by an arc equal in length to the radius of the circle. The radian measure of any angle is expressed as the ratio of the arc that the angle, with its vertex at the center of the circle, subtends to the radius of the circle. This ratio is constant, for a fixed angle, for all circles. The radian measure of an angle is not the ratio of the length of the chord subtended by the angle to the radius of the circle but the ratio of the length of the arc to the radius.

It can be seen that the radian measure of an angle and the so-called degree measure are related, as the circumference of a circle is given by the formula

C = 2pr

in which r is the radius of the circle and p is the number 3.14159.... Because the circumference of a circle contains exactly 2p radii, and because an arc of length r subtends one radian, it follows that

2p radians = 360 degrees

Division of 360° by 2 shows that one radian is about equal to 57°17’44.8”. For most practical applications, the following approximations are sufficiently accurate:

one radian = 57.3 degrees

one degree = 0.01745 radians

The degree and the radian are angular units of different size and may be used interchangeably. The degree is used more frequently by engineers and technicians, and radian measure is used almost exclusively in theoretical studies such as calculus because of the greater simplicity of certain results, notably the derivative and infinite-series expansion of the trigonometric functions. It should be noted that while the symbol ° is used to denote degree, no symbol is normally used to denote radian measure.