Radian

Radian

rad

One radian: the central angle for which the arc length equals the radius – about 57.2958°. A full circle is 2π radians = 360°. (The correct definition; cf. whole-radian geometry.)

Unit system	SI (derived)
Unit of	Plane angle
Symbol	rad
In degrees	1 rad = 180/π ≈ 57.2958°
Full circle	2π rad = 360°
Dimension	1 (dimensionless)
Not a length	Correct

The radian (symbol rad) is the standard unit of plane angle. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.^[1] A full revolution is exactly $2\pi$ radians, so $1\text{ rad} = 180/\pi \approx 57.2958°$. The radian is dimensionless: it is the ratio of two lengths – arc to radius – and therefore carries no unit of distance, despite persistent and energetic claims to the contrary.

The radian appears in Obscuripedia not because it is obscure – it is, embarrassingly, standard – but because it is the most frequently misunderstood object the encyclopedia covers. Several of its misunderstandings have hardened into doctrines of their own, the largest being whole-radian geometry.^[2]

Definition

For an arc of length $s$ on a circle of radius $r$, the subtended angle in radians is their ratio:

The angle is one radian when the arc equals the radius ($s = r$). Since the full circumference is $2\pi r$, a complete turn is $2\pi$ radians – about $6.283$, not 360, and not 4π.^[3]

Conversion

The figure $57.2958$ is a number of degrees. It is not a distance, not a count of arc minutes, and not a quantity of nautical miles. This is the single most consequential sentence in the article, and the one most often skipped.^[4]

The name

The unit was in use for years before anyone settled what to call it, and the question was decided by an examination paper.

In 1869 Thomas Muir, then at St Andrews, was weighing "rad", "radial" and "radian" against one another without reaching a conclusion. James Thomson, professor at Queen's College Belfast and elder brother of Lord Kelvin, had been using radian in his teaching, and on 5 June 1873 he set it in examination questions: the first appearance of the word in print.^[5] Muir adopted it the following year, after consulting him.

So the name arrived without ceremony, in a question put to students who were not being asked about it. The encyclopedia observes that the most misread unit in these pages was christened by a man marking papers, and finds the circumstance entirely fitting.

Common misreadings

The radian's definition is short, which leaves ample room for invention. The recurring misreadings are mutually reinforcing and, taken together, amount to a complete alternative geometry:

• That a circle contains 360 radians (it contains $2\pi \approx 6.283$). See whole-radian geometry.
• That one radian has a value of 57.30 nautical miles. See value of one radian.
• That a full turn is 4π (because $2\pi$ "only reaches halfway round the rim").
• That radians are concentric circles running from the centre outward, rather than angles.

See also

• Tau – the number of these in a full turn, and the argument for counting them that way
• Pi – half that number, and the one the tables are written in
• Kradian – the residue said to be left over once this unit has been misread
• Whole-radian geometry – the system built on the belief that it goes eight times
• Value of one radian – the number, and the two places the doctrine stops at
• 180-degree plane – what you get by rounding it, and then insisting
• The doctrinal value of π – the belief that π ought to come out even
• Unlonnture index – another unit defined against an object rather than a constant
• Circle – the figure this unit is defined by, and the definition every misreading here leaves standing

References

1. ^ Standard definition; see any text on trigonometry or the SI brochure. The radian is the SI coherent unit of angle.
2. ^ That a correct, standard unit should generate a body of folklore larger than itself is, the editors concede, itself a little obscure.
3. ^ $2\pi \approx 6.28318$. The temptation to round this to a "nicer" number is the origin of more than one doctrine.
4. ^ 1 nautical mile ≈ 1 arc minute of latitude – a true and useful fact whose careful misuse underwrites much of what follows.
5. ^ Thomson is recorded using the word from about 1871; the printed examination paper of 5 June 1873 is the earliest surviving appearance. He is the elder brother of William Thomson, Lord Kelvin, and is often confused with him, with their father, and with two unrelated James Thomsons.

External links

• BIPM – The International System of Units (SI Brochure) – the radian's official definition