Turn (angle)

Turn
Unit of	Plane angle
Symbol	tr, pla, τ
Conversions
1 tr in ...	... is equal to ...
radians	2π rad ≈ 6.283185307... rad
milliradians	2000π mrad ≈ 6283.185307... mrad
degrees	360°
gradians	400g

Counterclockwise rotations about the center point where a complete rotation is equal to 1 turn.

A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc. or cyl.), revolution (abbreviated rev.), complete rotation (abbreviated rot.) or full circle. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, points, etc.

Subdivisions

A turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a "percentage protractor".

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points. The binary degree, also known as the binary radian (or brad), is 1/256 turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2ⁿ equal parts for other values of n.

The notion of turn is commonly used for planar rotations.

History

The word turn originates via Latin and French from the Greek word τόρνος (tórnos – a lathe).

In 1697, David Gregory used π/ρ (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius. However, earlier in 1647, William Oughtred had used δ/π (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol π on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones. Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

The Latin word for turn is versor, which represents a rotation about an arbitrary axis in three-dimensional space. Versors form points in elliptic space and motivate the study of quaternions, an algebra developed by W. R. Hamilton in the 1840s.

Percentage protractors have existed since 1922, but the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962. Some measurement devices for artillery and satellite watching carry milliturn scales.

Unit symbols

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: plenus angulus 'full angle') for turns. Covered in DIN 1301-1 (October 2010), the so-called Vollwinkel ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU and Switzerland.

The standard ISO 80000-3:2006 mentions that the unit name revolution with symbol "r" is used with rotating machines, as well as using the term turn to mean a full rotation. The standard IEEE 260.1:2004 also uses the unit name rotation and symbol "r".

The scientific calculators HP 39gII and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to newRPL for the HP 50g in 2016, and for the hp 39g+, HP 49g+, HP 39gs, and HP 40gs in 2017. An angular mode TURN was suggested for the WP 43S as well, but the calculator instead implements "MULπ" (multiples of π) as mode and unit since 2019.

Unit conversion

The circumference of the unit circle (whose radius is one) is 2π.

A comparison of angles expressed in degrees and radians.

One turn is equal to 2π (≈ 6.283185307179586) radians, 360 degrees, or 400 gradians.

Proposals for a single letter to represent 2π

See also: Pi § Adoption of the symbol π

An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which is expressed here using the Greek letter tau (τ).

In 1746, Leonard Euler first used the Greek letter pi to represent the circumference divided by the radius (i.e. Pi is approx. 6.28...) of a circle.

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. His proposal used a "π with three legs" symbol to denote the constant ($\pi \!\;\!\!\!\pi =2\pi$).

In 2008, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π.

The Greek letter theta derives from the Phoenician and Hebrew letter teth, 𐤈 or ט, and it has been observed that the older version of the symbol, which means wheel, resembles a wheel with four spokes. It has also been proposed to use the wheel symbol, teth, to represent the quantity 2π, and more recently a connection has been made among other ancient cultures on the existence of a wheel, sun, circle, or disk symbol—i.e. other variations of teth—as representation for 2π.

In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: τ = 2π. He offered two reasons. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as 3τ/4 rad instead of 3π/2 rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where τ is used instead of π.

Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities. However, the use of τ has become more widespread, for example:

• In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.
• The τ-functionality is made available in the Google calculator and in several programming languages like Python, Raku, Processing, Nim, Rust, Java and .NET.
• It has also been used in at least one mathematical research article, authored by the τ-promoter Peter Harremoës.

The following table shows how various identities appear if τ := 2π was used instead of π. For a more complete list, see List of formulae involving π.

Formula	Using π	Using τ	Notes
1/4 of a circle	π/2 rad	τ/4 rad	τ/4 rad is a quarter of a circle and a quarter of τ
Circumference C of a circle of radius r	C = 2πr	C = τr
Area of a circle	A = πr2	A = τr2/2	Recall that the area of a sector of angle θ (measured in radians) is A = θr2/2. Also, the area of a circle with radius = r and circumference = C is equal to half the area of a rectangle with height = r and width = C, which is equal to the area of a right triangle with Opposite = r and Adjacent = C.
Area of a regular n-gon with unit circumradius	A = n/2 sin 2π/n	A = n/2 sin τ/n
Volume of an n-ball	$V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}$	$V_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{n!!}}(1+n\operatorname {mod} 2)R^{n}$
Surface area of an n-ball	$S_{n}(R)={\frac {2\pi ^{\frac {n+1}{2}}}{\Gamma {\big (}{\frac {n+1}{2}}{\big )}}}R^{n}$	$S_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n+1}{2}}\right\rfloor }}{(n-1)!!}}(2-(n\operatorname {mod} 2))R^{n}$
Cauchy's integral formula	$f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$f(a)={\frac {1}{\tau i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$
Standard normal distribution	$\varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}$	$\varphi (x)={\frac {1}{\sqrt {\tau }}}e^{-{\frac {x^{2}}{2}}}$
Stirling's approximation	$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$	$n!\sim {\sqrt {\tau n}}\left({\frac {n}{e}}\right)^{n}$
Euler's identity	0       eiπ = − 1 eiπ + 1 = 0	0     eiτ = 1 eiτ − 1 = 0
nth roots of unity	$e^{2\pi i{\frac {k}{n}}}=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}}$	$e^{\tau i{\frac {k}{n}}}=\cos {\frac {k\tau }{n}}+i\sin {\frac {k\tau }{n}}$
Reduced Planck constant	$\hbar ={\frac {h}{2\pi }}$	$\hbar ={\frac {h}{\tau }}$	h is the Planck constant.
Angular frequency	$\omega ={{2\pi } \over T}={2\pi f}$	$\omega ={{\tau } \over T}={\tau f}$
Reactance of an inductor	2πfL	τfL
Susceptance of a capacitor	2πfC	τfC

Examples of use

• As an angular unit, the turn or revolution is particularly useful for large angles, such as in connection with electromagnetic coils and rotating objects. See also winding number.
• The angular speed of rotating machinery, such as automobile engines, is commonly measured in revolutions per minute (RPM).
• Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.

See also

• Ampere-turn
• Hertz (modern) or Cycle per second (older)
• Angle of rotation
• Revolutions per minute
• Repeating circle
• Spat (unit) — the solid angle counterpart of the turn, equivalent to 4π steradians.
• Unit interval
• Spread (rational trigonometry)
• Modulo operation
• Twist (mathematics)

External links

• Tau manifesto