Radian (rad)
Definition, derivation and practical use in the International System of Units
Plane angle states how far two lines lean towards each other, or how far something has turned about a point. In everyday life angles are measured in degrees, but in mathematics, physics and programming the radian (rad) is the unit that counts. The radian is one of the derived SI units and the only angle unit the SI itself defines.
The radian is unusual among SI units because it is dimensionless: it is the ratio of two lengths. As for the meter and the kilogram, it is still necessary to distinguish between definition, realization and practical use.
The definition of the radian in the SI system
Within the International System of Units (SI), the radian is defined as:
the plane angle between two radii of a circle that cuts off on the circumference an arc equal in length to the radius.
In formula form: 1 rad = 1 m/m = 1. An angle in radians is the arc length divided by the radius, θ = s/r. The definition rests on:
- the meter, because both the arc length and the radius are lengths,
- the geometry of the circle, where one full turn gives the arc length 2πr and therefore the angle 2π rad.
Since one full turn is both 2π rad and 360°, 1 rad = 180°/π ≈ 57.29578° and 1° = π/180 rad ≈ 0.017453 rad.
Fact box: the history of the radian
| Year | Event |
| 1714 | Roger Cotes describes the angle as an arc divided by its radius, the idea behind the radian. |
| 1873 | The word radian first appears in print, in examination questions set by James Thomson at Queen’s College, Belfast. |
| 1960 | The 11th General Conference on Weights and Measures (CGPM) establishes the SI and puts the radian and the steradian in a class of their own, the supplementary units. |
| 1980 | The International Committee for Weights and Measures (CIPM) interprets the supplementary units as dimensionless derived units. |
| 1995 | The 20th CGPM abolishes the class of supplementary units. The radian becomes a derived SI unit with a special name. |
How big is a radian?
One radian is about 57.3 degrees, a little less than one sixth of a turn. A right angle is π/2 ≈ 1.5708 rad, half a turn is π ≈ 3.1416 rad and a full turn is 2π ≈ 6.2832 rad. A full turn therefore holds just over six radians.
Small angles are often given in milliradians. A milliradian (1 mrad = 0.001 rad) is roughly the angle that one meter takes up at a distance of one kilometer. The NATO mil divides the turn into 6,400 parts and is close to a milliradian, while a full turn holds just over 6,283 milliradians.
Definition, realization and use
- The definition states what a radian is in principle: an arc as long as the radius.
- The realization refers to the angle-measuring instruments and angle standards with which metrology institutes and calibration laboratories measure angles traceably.
- The use refers to trigonometry, rotational motion, oscillations, surveying and programming, where almost every mathematical function works in radians.
Mathematics needs the radian because it keeps the formulas simple. The derivative of sin x is cos x only when x is in radians, and for small angles sin θ ≈ θ.
How the radian is realized
The radian has no physical prototype and no constant of nature in its definition. Angle measurement has instead a property no other quantity has: a full turn is always exactly 2π rad. By dividing the circle and comparing the parts with each other, a method known as circle closure, an angle-measuring instrument can be calibrated without any external reference.
National metrology institutes use angle comparators with rotary tables, optical polygons and autocollimators. With them they reach uncertainties far below one arcsecond. Calibration laboratories for rotary encoders, theodolites and spirit levels trace their measurements to these standards.
Practical measurement and sources of error
In practice, angles are measured with protractors, rotary encoders, theodolites and inclinometers. The most common error is still not the measurement but the conversion:
- The wrong calculator mode: in DEG mode a calculator reads sin(1) as the sine of one degree, in RAD mode as the sine of one radian. The answers are 0.017452 and 0.841471.
- A rounded π: with π = 3.14 a full turn becomes 6.28 rad instead of 6.283185 rad. That is an error of about 0.05 percent.
- Different mils: the NATO mil is 1/6,400 of a turn, the old Swedish streck 1/6,300 of a turn and the milliradian 1/1,000 rad. They are almost the same size but not interchangeable.
The radian and other units
The radian links angle to several other quantities in the SI:
| Unit | Quantity | Relation to the radian |
| meter (m) | Arc length | s = r · θ |
| hertz (Hz) | Angular frequency | ω = 2π · f, in rad/s |
| joule (J) | Work done by turning | W = M · θ |
| newton (N) | Torque | M = F · r, in newton-meters |
| steradian (sr) | Solid angle | The three-dimensional counterpart, 1 sr = 1 m²/m² |
| candela (cd) | Luminous intensity | Defined per steradian |
Radians to degrees and other conversions
To convert radians to degrees, multiply by 180/π. To convert degrees to radians, multiply by π/180. So 2 rad is 114.59156°, and 45° is 0.785398 rad. The radian to degree converter and the reverse degree to radian converter show the formula and a full table.
The angle converter also converts between turns, degrees, arcminutes, arcseconds, gradians and mils. For schools there is an editable angle conversion chart, and for rotation the frequency converter converts between revolutions per minute and hertz.
Summary
The radian is the SI unit of plane angle and is defined as the angle whose arc is as long as the radius. A full turn is 2π rad, and one radian is about 57.3 degrees.
The unit has no dimension, but it does a great deal of work. The radian keeps the formulas of trigonometry simple, gives angular velocity in rad/s and is what computer programs calculate with, even when the answer is finally shown in degrees.