Turn (angle)

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Unit ofPlane angle
Symboltr, pla or τ
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.), revolution (abbreviated rev.), complete rotation (abbreviated rot.) or full circle.

Subdivisions of a turn include half-turns, quarter-turns, centi-turns, milli-turns, points, etc.


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″.[1][2] 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.[3] 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.[4]

The notion of turn is commonly used for planar rotations.


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.[5][6] 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.[7] Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922,[8] but the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962.[1][2] Some measurement devices for artillery and satellite watching carry milliturn scales.[9][10]

Unit symbols[edit]

The German standard DIN 1315 (March 1974) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.[11][12] Covered in DIN 1301-1 (October 2010), the so called Vollwinkel (English: "full angle") is not an SI unit. However, it is a legal unit of measurement in the EU[13][14] and Switzerland.[15]

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.[16][17] An angular mode TURN was suggested for the WP 43S as well,[18] but the calculator instead implements MULπ (multiples of π) as mode and unit since 2019.[19][20]

Unit conversion[edit]

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

One turn is equal to 2π (≈ 6.283185307179586)[21] radians.

Conversion of common angles
Turns Radians Degrees Gradians, or gons
0 0 0g
1/24 π/12 15° 16+2/3g
1/12 π/6 30° 33+1/3g
1/10 π/5 36° 40g
1/8 π/4 45° 50g
1/2π 1 c. 57.3° c. 63.7g
1/6 π/3 60° 66+2/3g
1/5 2π/5 72° 80g
1/4 π/2 90° 100g
1/3 2π/3 120° 133+1/3g
2/5 4π/5 144° 160g
1/2 π 180° 200g
3/4 3π/2 270° 300g
1 2π 360° 400g

Tau proposals[edit]

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 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 ().[22]

In 2010, Michael Hartl proposed to use tau to represent Palais' 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.[23] Hartl's Tau Manifesto[24] gives many examples of formulas that are asserted to be clearer where τ is used instead of π.[25][26][27]

Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities.[28] However, the use of τ has become more widespread,[29] for example:

  • In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.[30]
  • In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.[31]
  • The τ-functionality is made available in the Google calculator and in several programming languages like Python,[32] Raku,[33] Processing,[34] Nim,[35] and Rust.[36]
  • It has also been used in at least one mathematical research article,[37] authored by the τ-promoter Peter Harremoës.[38]
  • In 2020, for release 5.0, Tau was added to .NET Core (which is being rebranded as ".NET" for the 5.0 release).[39]

The following table shows how various identities and inequalities appear if τ := 2π was used instead of π.[40][41]

Using τ := 2π Using π Formula Notes
τ/4 π/2 1/4 of a circle (as an angle in radians)
C = τr C = 2πr Circumference C of a circle of radius r
A = τr2/2 A = πr2 Area of a circle
  • Recall the area of a sector of angle θ (measured in radians) is A = θr2/2.
  • The 1/2 more clearly expresses that area is the integral of circumference.
  • Compare to Kinetic energy mv2/2 and spring energy kx2/2.
A = n/2 sin τ/n A = n/2 sin /n Area of a regular n-gon with unit circumradius
Volume of an n-ball
Surface area of an n-ball
Cauchy's integral formula
Standard normal distribution
Stirling's approximation
0     e = 1
e - 1 = 0
0      e = - 1
e + 1 = 0
Euler's identity
nth roots of unity
Reduced Planck constant
  • h is the natural unit of electromagnetic action.
Angular frequency
τfL 2πfL Reactance of an inductor
τfC 2πfC Susceptance of a capacitor

Examples of use[edit]

  • 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 or RPM.
  • Turn is used in complex dynamics for measure of external and internal angles. The sum of external angles of a polygon equals one turn. Angle doubling map is used.
  • Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.[8]

Kinematics of turns[edit]

In kinematics, a turn is a rotation less than a full revolution. A turn may be represented in a mathematical model that uses expressions of complex numbers or quaternions. In the complex plane every non-zero number has a polar coordinate expression z = r cis a = r(cos a + i sin a) where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + iy by an element u = exp(bi) that lies on the unit circle:

Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley.[42]

The Latin term for turn is versor, which is a quaternion that can be visualized as an arc of a great circle. The product of two versors can be compared to a spherical triangle where two sides add to the third. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation. This algebraic expression of rotation was initiated by William Rowan Hamilton in the 1840s (using the term versor), and is a recurrent theme in the works of Narasimhaiengar Mukunda as "Hamilton's theory of turns".

See also[edit]


  1. ^ a b Hoyle, Fred (1962). Chandler, M. H. (ed.). Astronomy (1 ed.). London, UK: Macdonald. LCCN 62065943. OCLC 7419446. (320 pages)
  2. ^ a b Klein, Herbert Arthur (2012) [1988, 1974]. "Chapter 8: Keeping Track of Time". The Science of Measurement: A Historical Survey (The World of Measurements: Masterpieces, Mysteries and Muddles of Metrology). Dover Books on Mathematics (corrected reprint of original ed.). Dover Publications, Inc. / Courier Corporation (originally by Simon & Schuster, Inc.). p. 102. ISBN 978-0-48614497-9. LCCN 88-25858. Retrieved 2019-08-06. (736 pages)
  3. ^ "ooPIC Programmer's Guide - Chapter 15: URCP". ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 2008-06-28. Retrieved 2019-08-05.
  4. ^ Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com. Archived from the original on 2019-06-30. Retrieved 2019-08-05.
  5. ^ Beckmann, Petr (1989) [1970]. A History of Pi. Barnes & Noble Publishing.
  6. ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. The Mathematical Association of America. p. 165.
  7. ^ Veling, Anne (2001). "Pi through the ages". veling.nl. Archived from the original on 2009-07-02.
  8. ^ a b Croxton, Frederick E. (1922). "A Percentage Protractor - Designed for Use in the Construction of Circle Charts or "Pie Diagrams"". Journal of the American Statistical Association. Short Note. 18 (137): 108–109. doi:10.1080/01621459.1922.10502455.
  9. ^ Schiffner, Friedrich (1965). "Bestimmung von Satellitenbahnen". Mitteilungen der Uraniasternwarte (in German). Wien.
  10. ^ Hayes, Eugene Nelson (1975) [1968]. Trackers of the Skies. History of the Smithsonian Satellite-tracking Program. Cambridge, Massachusetts, USA: Academic Press / Howard A. Doyle Publishing Company.
  11. ^ German, Sigmar; Drath, Peter (2013-03-13) [1979]. Handbuch SI-Einheiten: Definition, Realisierung, Bewahrung und Weitergabe der SI-Einheiten, Grundlagen der Präzisionsmeßtechnik (in German) (1 ed.). Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, reprint: Springer-Verlag. p. 421. ISBN 978-3-32283606-9. 978-3-528-08441-7, 978-3-32283606-9. Retrieved 2015-08-14.
  12. ^ Kurzweil, Peter (2013-03-09) [1999]. Das Vieweg Einheiten-Lexikon: Formeln und Begriffe aus Physik, Chemie und Technik (in German) (1 ed.). Vieweg, reprint: Springer-Verlag. p. 403. doi:10.1007/978-3-322-92920-4. ISBN 978-3-32292920-4. 978-3-322-92921-1. Retrieved 2015-08-14.
  13. ^ "Richtlinie 80/181/EWG - Richtlinie des Rates vom 20. Dezember 1979 zur Angleichung der Rechtsvorschriften der Mitgliedstaaten über die Einheiten im Meßwesen und zur Aufhebung der Richtlinie 71/354/EWG" (in German). 1980-02-15. Archived from the original on 2019-06-22. Retrieved 2019-08-06.
  14. ^ "Richtlinie 2009/3/EG des Europäischen Parlaments und des Rates vom 11. März 2009 zur Änderung der Richtlinie 80/181/EWG des Rates zur Angleichung der Rechtsvorschriften der Mitgliedstaaten über die Einheiten im Messwesen (Text von Bedeutung für den EWR)" (in German). 2009-03-11. Archived from the original on 2019-08-06. Retrieved 2019-08-06.
  15. ^ "Art. 15 Einheiten in Form von nichtdezimalen Vielfachen oder Teilen von SI-Einheiten". Einheitenverordnung. Der Bundesrat - Das Portal der Schweizer Regierung (in German). Schweizerischer Bundesrat. 1994-11-23. 941.202. Archived from the original on 2019-05-10. Retrieved 2013-01-01.
  16. ^ Lapilli, Claudio Daniel (2016-05-11). "RE: newRPL: Handling of units". HP Museum. Archived from the original on 2017-08-10. Retrieved 2019-08-05.
  17. ^ Lapilli, Claudio Daniel (2018-10-25). "Chapter 3: Units - Available Units - Angles". newRPL User Manual. hpgcc3. Archived from the original on 2019-08-06. Retrieved 2019-08-07.
  18. ^ Paul, Matthias R. (2016-01-11). "RE: WP-32S in 2016?". HP Museum. Archived from the original on 2019-08-05. Retrieved 2019-08-05.
  19. ^ Bonin, Walter (2019) [2015]. WP 43S Owner's Manual (PDF). 0.12 (draft ed.). pp. 72, 118–119, 311. ISBN 978-1-72950098-9. Retrieved 2019-08-05. [1] [2] (314 pages)
  20. ^ Bonin, Walter (2019) [2015]. WP 43S Reference Manual (PDF). 0.12 (draft ed.). pp. iii, 54, 97, 128, 144, 193, 195. ISBN 978-1-72950106-1. Retrieved 2019-08-05. [3] [4] (271 pages)
  21. ^ Sequence OEISA019692
  22. ^ Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. New York, USA: Springer-Verlag. 23 (3): 7–8. doi:10.1007/bf03026846. S2CID 120965049. Archived (PDF) from the original on 2019-07-18. Retrieved 2019-08-05.
  23. ^ Hartl, Michael (2019-03-14) [2010-03-14]. "The Tau Manifesto". Archived from the original on 2019-06-28. Retrieved 2013-09-14.
  24. ^ Hartl, Michael (2010-03-14). "The Tau Manifesto" (PDF). Archived (PDF) from the original on 2019-07-18. Retrieved 2019-08-05.
  25. ^ Aron, Jacob (2011-01-08). "Michael Hartl: It's time to kill off pi". New Scientist. Interview. 209 (2794): 23. Bibcode:2011NewSc.209...23A. doi:10.1016/S0262-4079(11)60036-5.
  26. ^ Landau, Elizabeth (2011-03-14). "On Pi Day, is 'pi' under attack?". cnn.com. CNN. Archived from the original on 2018-12-19. Retrieved 2019-08-05.
  27. ^ Bartholomew, Randyn Charles (2014-06-25). "Let's Use Tau--It's Easier Than Pi - A growing movement argues that killing pi would make mathematics simpler, easier and even more beautiful". Scientific American. Archived from the original on 2019-06-18. Retrieved 2015-03-20.
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  30. ^ "Happy Tau Day!". blog.khanacademy.org. Retrieved 2020-12-19.
  31. ^ Coghlan, Nick (2017-02-25). "PEP 628 -- Add math.tau". Python.org. Archived from the original on 2019-07-22. Retrieved 2019-08-05.
  32. ^ "math — Mathematical functions". Python 3.7.0 documentation. Archived from the original on 2019-07-29. Retrieved 2019-08-05.
  33. ^ "Perl 6 terms". Archived from the original on 2019-07-22. Retrieved 2019-08-05.
  34. ^ "TAU". Processing. Archived from the original on 2019-07-22. Retrieved 2019-08-05.
  35. ^ "math". Nim. Archived from the original on 2019-07-22. Retrieved 2019-08-05.
  36. ^ "std::f64::consts::TAU - Rust". doc.rust-lang.org. Retrieved 2020-10-09.
  37. ^ Harremoës, Peter (2017). "Bounds on tail probabilities for negative binomial distributions". Kybernetika. 52 (6): 943–966. arXiv:1601.05179. doi:10.14736/kyb-2016-6-0943. S2CID 119126029.
  38. ^ Harremoës, Peter (2018-11-17). "Al-Kashi's constant τ" (PDF). Archived (PDF) from the original on 2019-07-22. Retrieved 2018-09-20.
  39. ^ https://github.com/dotnet/runtime/pull/37517
  40. ^ Abbott, Stephen (April 2012). "My Conversion to Tauism" (PDF). Math Horizons. 19 (4): 34. doi:10.4169/mathhorizons.19.4.34. S2CID 126179022. Archived (PDF) from the original on 2013-09-28.
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  42. ^ Morley, Frank; Morley, Frank Vigor (2014) [1933]. Inversive Geometry. Boston, USA; New York, USA: Ginn and Company, reprint: Courier Corporation, Dover Publications. ISBN 978-0-486-49339-8. Retrieved 2015-10-17.

External links[edit]