Degree of arc
Non-SI unit accepted for use · quantity: plane angle
01 · Definition
Three hundred and sixty is the only arbitrary number in all of metrology that survives every reform. The reason lies not in astronomy but in divisors.
A degree is one three-hundred-sixtieth of a full turn, that is π/180 of a radian. It is not divided decimally: sixty arcminutes to a degree, sixty arcseconds to a minute — an inheritance of the Babylonian count by sixties that reached us through Ptolemy. The degree is not part of the SI but is accepted for use, and nowhere outside mathematics has the radian managed to displace it: maps, drawings, navigation and every angle-measuring instrument carry degrees. The number 360 divides without remainder by 24 different numbers, including every one from one to six — try dividing 400 or 100 that way.
Sixty arcminutes to a degree is no whim but a convenience of division: sixty divides by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. Babylonian tables of reciprocals came out short, decimal ones would have given endless fractions, and the whole of Ptolemy’s trigonometry rests on exactly this count.
02 · Conversion
Enter an angle and the sheet lays it out across the scales
SI prefixes are not applied to the degree: instead of milli- and microdegrees the angle is broken into minutes and seconds, and in astrometry one moves to milliarcseconds — there the prefix belongs to the second, not to the degree.
An arcsecond is very nearly five millimetres over a kilometre. It is from this that the astronomical unit of distance grew: the distance at which the radius of Earth’s orbit subtends one second was named the parsec.
| Unit | Name | Value | Where it appears |
|---|---|---|---|
| {u} | {name} | {value} | {note} |
03 · Orders of magnitude
from a microarcsecond to a full turn04 · Measuring instruments
What angles are taken with
Horizontal and vertical limbs with marks every minute and an optical micrometer for the seconds. Precise models take an angle to within half an arcsecond.
Double reflection lets one bring a star down to the horizon and take its altitude straight from a rolling deck. The limb spans 60°, which is where the instrument gets its name.
On the shop floor and the building site angles are read to a tenth of a degree: a mechanical scale, an accelerometer inclinometer, or a laser level with an angular sweep.
Gaia measures stellar positions to about twenty microarcseconds — the angle a human hair subtends from a thousand kilometres away.
05 · Writing rules
The angle takes no space, the temperature does
The degree-of-arc sign is set tight against the number: 25°. The degree Celsius is a unit of another kind, and there the space is obligatory: 25 °C. Arcminutes and arcseconds are marked with a prime and a double prime, not with an apostrophe and quotation marks. Inside sexagesimal notation there are no spaces: 41°24′12″. A fractional part is allowed only in the last place.
Coordinates are written either sexagesimally or in signed decimal degrees: 55.7558° and 37.6173° for Moscow, minus for southern latitude and western longitude. Mixing the forms in one line is not allowed — it is the commonest source of error in geodata.
06 · Neighbouring units
Plane angle is measured on three scales: the radian rules in mathematics, the degree in practice, and the arcminute and arcsecond divide the degree where a decimal fraction is awkward. All three are dimensionless and convert into one another by factors.
One arcminute along the meridian is the nautical mile, 1852 metres by definition. Marine navigation and the grid of degrees hold onto each other through precisely this coincidence.
07 · Historical section
Those who tried to divide the turn otherwise
The metric version of the angle: 400 gradians to a turn, exactly 100 to a right angle. It survived the reform only in French geodesy and in the GRA key on calculators.
The angle a metre subtends from a kilometre away. NATO divides the turn into 6400 such units, the Soviet tradition into 6000, and their sight settings do not agree.
The compass rose divides the horizon into 32 parts. A course was named in words — «north-north-east» — and that was safer than reading figures in the dark on a rolling ship.
The sky turns, so right ascension is measured in time: 24 hours to a full revolution. Star catalogues are still printed in hours, minutes and seconds of time.
The number 360 holds not by tradition but by divisibility
The metric reform replaced almost everything: cubits by metres, pounds by kilograms — but the angle it could not take. The gradian with its four hundred parts was honestly introduced by decree, and it remained only in two or three national surveying services. The reason is simple: 360 divides by 24 different numbers, among them every one from one to six, and eight, nine, ten, twelve and fifteen. A third of a turn is exactly 120°, a sixth exactly 60°, while in gradians the same parts give 133.33 and 66.67. For a drawing, a map and an instrument limb that settles everything. The radian is mathematically impeccable, but one cannot draw an angle of 0.5236 radian without a calculator, whereas thirty degrees is drawn by any protractor.
Catalogue · units of measurement
A passport for every quantity
Seven SI base units, twenty-two derived ones with special names, and the non-SI units that neither engineering nor everyday life can do without. Each gets its own sheet: definition, conversion, instruments, rules of writing. In 36 languages.
Non-SI units
open the degree data sheetSI base units
an angle is dimensionless — it contains no base unitsSI derived units with names of their own
in terracotta — the SI units of angleSheet language
36 languages. The symbol dB and the formulas are not translated — they are the same in every locale; what is translated is the prose, the local land measures and the writing rules.