Radian
SI derived unit · quantity: plane angle
01 · Definition
The radian is the only SI unit that equals one. Its dimension is a metre per metre, that is, no dimension at all — and yet it has a name, a symbol and a place in the table of derived units.
One radian is the angle at which the length of the arc equals the radius. Hence its awkward look in degrees: 57 degrees 17 minutes 45 seconds. In return, all of mathematics gets simpler: the derivative of the sine equals the cosine only if the argument is in radians, arc length is the angle times the radius, and angular velocity is measured in radians per second with no conversion factors at all. A full turn is 2π radians, and that is the only reason π appears in trigonometry on every line.
Angular velocity is measured in radians per second, frequency in hertz, and the two are linked by a factor of 2π. Writing ω in hertz is formally possible, since the radian is dimensionless, but the SI brochure does not allow it: the quantities are different.
02 · Conversion
Enter an angle — the sheet will convert it
The degree survived the metric reform: the attempt to introduce the grad, where a right angle equals a hundred units, failed everywhere except French surveying. Three hundred and sixty divides by two, three, four, five, six and nine without a remainder — that convenience turned out to be stronger than decimal logic.
The milliradian is handy in practice: an angle of one milliradian gives a displacement of one metre at a kilometre of distance. Snipers and artillerymen count in mils for that very reason, and the arithmetic in the head comes down to division.
| Unit | Name | Value | Where it is met |
|---|---|---|---|
| rad | radian, the SI unit | 1 | mathematics, physics, engineering |
| mrad | milliradian | 1000 | sights, lasers, optics |
| µrad | microradian | 1 000 000 | alignment, autocollimator |
| nrad | nanoradian | 1·10⁹ | interferometry |
| turn | turn, 2π radians | 0.15915 | rotation, windings |
| m/m | a metre per metre — that is exactly 1 | 1 | the radian is dimensionless |
| ° | degree, 1/360 of a circle | 57.3 | everyday life, surveying, navigation |
| ′ | minute of arc, 1/60 of a degree | 3438 | sea charts, astronomy |
| ″ | second of arc | 206 265 | theodolite, astrometry |
| mas | millisecond of arc | 2.063·10⁸ | stellar parallaxes |
| grad | grad (gon), 1/400 of a circle | 63.66 | French surveying |
| mil | mil, 1/6400 of a turn | 1019 | artillery |
| rad | radian | 1 | the present-day unit |
The degree, the minute and the second of arc remain in the SI as non-system units — neither navigation nor astronomy can do without them. The grad appeared during the French metric reform: a quarter circle in a hundred parts, but it took root only in surveying.
03 · Orders of magnitude
from the resolution of a telescope to a full turnOne radian — 57.3 degrees
04 · Measuring instruments
What angles are measured with
A divided circle with a reading microscope: the telescope is aimed at the target and the reading is taken to within a second of arc. The surveyor speaks in degrees, minutes and seconds, but the built-in computer converts them into radians, because otherwise the trigonometry will not work.
A slotted disc on the shaft: a photodetector counts pulses and divides the turn into thousands of counts. A robot controller stores the joint position directly in radians — that way trigonometry and derivatives are computed without a single coefficient.
It sends a parallel beam onto a mirror and catches the reflection: a tilt of a few microradians shifts the mark by fractions of a millimetre. At such angles the sine, the tangent and the angle itself agree to the sixth decimal, so the instrument simply divides the shift by the focal length.
Gaia measured the positions of stars to about twenty microseconds of arc — a tenth of a nanoradian, the thickness of a hair seen from two thousand kilometres. Angles there are computed only in radians: in degrees the numbers stop fitting into the number format.
05 · Writing rules
rad may be dropped, but the degree sign may not
Since the radian is dimensionless, «sin 2» and «sin 2 rad» mean the same thing, and in mathematical texts the symbol is dropped. But if the angle is in degrees, the sign is obligatory: sin 2° differs from sin 2 thirtyfold. The symbol rad is lower-case; in dosimetry the same «rad» is a unit of absorbed dose, so in Russian texts the angle is written «рад» only with a caveat.
The degree, the minute and the second of arc are not part of the SI but are permitted for use. Prefixes are not applied to them: there is no such thing as a millidegree, and fractions are written as minutes and seconds or as a decimal.
06 · Neighbouring units
The radian and the steradian are the two dimensionless SI units: the first for plane angle, the second for solid angle. Both equal one, both have their own name, and both cause the same headache for anyone writing dimension checks.
m / m
m² / m²
ω = 2πf
At angles below 0.1 radian the sine, the tangent and the angle itself agree to better than one per cent — that is what the whole optics of small deviations and the theory of the pendulum rest on.
07 · Historical section
How the circle was divided before the radian
A sexagesimal trace of Babylonian mathematics: it divides by two, three, four, five, six without a remainder. Ptolemy split the degree into sixty minutes and the minute into sixty seconds, and the notation has survived unchanged for eighteen centuries.
A metric attempt: a right angle is a hundred grads, a full circle four hundred. It survived in European surveying and in the calculators of that trade — the label GRA next to DEG and RAD.
A sighting unit: roughly a milliradian. In the Russian tradition the circle is divided into six thousand, in NATO into 6400. The convenience is the same: one mil is one metre at a kilometre.
The circle is divided into 24 hours by the daily rotation of the Earth. Right ascensions of stars are still published in hours, minutes and seconds: one hour is fifteen degrees.
A unit that cannot be removed and cannot be kept
Formally the radian equals one, so the International Committee for Weights and Measures argued over its status for a long time: in 1960 it was called a supplementary unit, in 1980 there was a proposal to recognise it as a dimensionless derived unit, and in 1995 a compromise was adopted — a derived unit with a special name and dimension one. The practical problem remained: in formulas such as ω = 2πf the radian appears and disappears, and dimension-checking systems in engineering software are forced to treat angle as an exception.
Catalogue · the International System of Units
A passport for every quantity
Seven base units, twenty-two derived ones with their own names, and an archive of systems that have gone out of use. Each has its own sheet: definition, dimension, conversion, instruments, rules of writing. In 36 languages.