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]

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

• Whole-radian geometry
• Value of one radian
• 180-degree plane
• Unlonnture index

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.