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HandWiki. Turn (Geometry). Encyclopedia. Available online: (accessed on 15 June 2024).
HandWiki. Turn (Geometry). Encyclopedia. Available at: Accessed June 15, 2024.
HandWiki. "Turn (Geometry)" Encyclopedia, (accessed June 15, 2024).
HandWiki. (2022, November 22). Turn (Geometry). In Encyclopedia.
HandWiki. "Turn (Geometry)." Encyclopedia. Web. 22 November, 2022.
Turn (Geometry)

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, centiturns, milliturns, points, etc.

angle measurement milliturns revolution

1. Subdivision of Turns

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 ​1256 turn.[1] 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.[2]

The notion of turn is commonly used for planar rotations.

2. 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.[3][4] 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.[5] Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922,[6] but the terms centiturns and milliturns were introduced much later by Fred Hoyle.[7]

The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.[8][9] Since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g.[10] In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.[11]

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.

3. Unit Conversion

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

One turn is equal to 2π (≈ 6.283185307179586)[12] 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

4. Tau Proposals

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 are expressed here using the Greek letter tau.

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 "pi with three legs" symbol to denote the constant ([math]\displaystyle{ \pi\!\;\!\!\!\pi }[/math] = 2π).[13]

In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: Template:Tau = 2π. He offered two reasons. First, Template:Tau 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 3/4Template:Tau rad instead of 3/2π rad. Second, Template:Tau visually resembles π, whose association with the circle constant is unavoidable.[14] Hartl's Tau Manifesto[15] gives many examples of formulas that are asserted to be clearer where tau is used instead of pi.[16][17][18]

The proposal is implemented in the Google calculator and in several computer programs like Python[19], Perl[20], Processing[21], and Nim[22]. It has also been used in at least one mathematical research article,[23] authored by the Template:Tau-promoter P. Harremoës.[24]

However, none of these proposals have received widespread acceptance by the mathematical and scientific communities.[25]

5. 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 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.

6. Kinematics of Turns

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) + ri 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 = ebi 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.[26]

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.


  1. "ooPIC Programmer's Guide". Archived from the original on 2008-06-28. 
  2. "Angles, integers, and modulo arithmetic". 
  3. A History of Pi. Barnes & Noble Publishing. 1989. 
  4. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. The Mathematical Association of America. 1994. p. 165. 
  5. "Pi through the ages". 
  6. "A Percentage Protractor". Journal of the American Statistical Association 18: 108–109. 1922. doi:10.1080/01621459.1922.10502455.
  7. Astronomy. London: Macdonald. 1962. 
  8. (in German) Handbuch SI-Einheiten: Definition, Realisierung, Bewahrung und Weitergabe der SI-Einheiten, Grundlagen der Präzisionsmeßtechnik (1 ed.). Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, reprint: Springer-Verlag. 2013-03-13. 978-3-528-08441-7, 9783322836069. ISBN 3322836061. Retrieved 2015-08-14. 
  9. (in German) Das Vieweg Einheiten-Lexikon: Formeln und Begriffe aus Physik, Chemie und Technik (1 ed.). Vieweg, reprint: Springer-Verlag. 2013-03-09. doi:10.1007/978-3-322-92920-4. 978-3-322-92921-1. ISBN 3322929205. Retrieved 2015-08-14. 
  12. Sequence OEIS: A019692
  13. "Pi is Wrong". The Mathematical Intelligencer (New York, USA: Springer-Verlag) 23 (3): 7–8. 2001. doi:10.1007/bf03026846. 
  14. "The Tau Manifesto". 2013-03-14. 
  16. "Interview: Michael Hartl: It's time to kill off pi". New Scientist 209 (2794): 23. 2011-01-08. doi:10.1016/S0262-4079(11)60036-5. Bibcode: 2011NewSc.209...23A.
  17. "On Pi Day, is 'pi' under attack?". 2011-03-14. 
  18. "Why Tau Trumps Pi". Scientific American. 2014-06-25. 
  19. "Python 3.7.0 documentation". 
  20. "Perl 6". 
  21. "Processing". 
  22. "Nim". 
  23. Harremoës, Peter. "Bounds on tail probabilities for negative binomial distributions". Kybernetika 52 (6): 943-966. doi:10.14736/kyb-2016-6-0943.
  24. Harremoës, Peter. "Al-Kashi's constant τ". Retrieved 20 September 2018. 
  25. "Life of pi in no danger – Experts cold-shoulder campaign to replace with tau". Telegraph India. 2011-06-30. Archived from the original on 13 July 2013. 
  26. Inversive Geometry. Boston, USA; New York, USA: Ginn and Company, reprint: Courier Corporation, Dover Publications. 2014. 0-486-49339-3. ISBN 978-0-486-49339-8. Retrieved 2015-10-17. 
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