radian

SI coherent derived unit with special name and symbol

Name	Symbol	Derived quantity	Expressed in terms of SI base units
radian	rad	plane angle	m/m
Definition The radian, symbol rad, is the SI coherent derived unit for plane angle and phase angle. One radian is defined as the angle subtended at the centre of a circle by an arc that is equal in length to the circle’s radius.

For a unit circle, with a radius of one metre, an angle of one radian at the circle’s centre subtends an arc on the circumference with a length of one metre.

The magnitude in radians of an angle θ subtended at the centre of a circle is equal to the ratio of the length of the arc s enclosed by the angle to the length of the circle’s radius r.

Conversely, the length s of the arc of a circle enclosed by an angle at the circle’s centre is equal to the product of the radius r of the circle and the magnitude in radians θ of the angle.

It follows that the magnitude in radians of one complete turn of a circle is equal to the length of the circumference, 2πr, divided by the length of the radius, r.

The phase angle (often just referred to as the “phase”) is the argument of any complex number. It is the angle between the positive real axis and the radius of the polar representation of the complex number in the complex plane.

Examples of angles

turns	radians in terms of τ (τ = 2π)	radians in terms of π	degrees
1⁄360	τ⁄360	π⁄180	1°
1⁄12	τ⁄12	π⁄6	30°
1⁄10	τ⁄10	π⁄5	36°
1⁄8	τ⁄8	π⁄4	45°
1⁄2π	1	1	≈ 57.3°
1⁄6	τ⁄6	π⁄3	60°
1⁄5	τ⁄5	2π⁄5	72°
1⁄4	τ⁄4	π⁄2	90°
1⁄3	τ⁄3	2π⁄3	120°
1⁄2	τ⁄2	π	180°
2⁄3	2τ⁄3	4π⁄3	240°
3⁄4	3τ⁄4	3π⁄2	270°
1	τ	2π	360°

Usage

The radian corresponds to the ratio of two lengths. However, the radian must only be used to express angles, and not to express ratios of lengths in general.

Degree

A historical convention for the measurement of plane angle is the non-SI unit degree. The conversion between radians and degrees follows from the relation: 1 turn = 360°.