View source for Turn (angle)

{{short description|Unit of plane angle where a full circle equals 1}}
{{Redirect2|360 degrees|360°}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
{{Use list-defined references|date=July 2022}}
{{Infobox unit
| name     = Turn
| othernames = Revolution, Cycles
| image = angle-fractions.png
| caption = {{longitem|Counterclockwise [[rotation]]s about the center point starting from the right, where a complete rotation corresponds to an angle of rotation of 1&nbsp;turn.}}
| standard = 
| quantity = [[Plane angle]]
| symbol   = tr
| symbol2  = pla
| symbol3  = rev
| symbol4  = cyc
| units1   = [[radian]]s
| inunits1 = {{math|2''π''}}&nbsp;rad<br/>≈ {{val|6.283185307|end=...|u=rad}}
| units3   = [[milliradian]]s
| inunits3 = {{math|2000''π''}}&nbsp;mrad<br/>≈ {{val|6283.185307|end=...|u=mrad}}
| units4   = [[Degree (angle)|degree]]s
| inunits4 = 360°
| units5   = [[gradian]]s
| inunits5 = 400<sup>g</sup>
}}

The '''turn''' (symbol '''tr''' or '''pla''') is a unit of [[plane angle]] measurement that is the measure of a [[complete angle]]—the angle [[Subtended angle|subtended]] by a complete [[circle]] at its center. One turn is equal to {{math|2[[Pi|''π'']]}}&nbsp;[[radian]]s, 360&nbsp;[[degree (angle)|degrees]] or 400&nbsp;[[gradian]]s. As an [[angular unit]], one turn also corresponds to one '''cycle''' (symbol '''cyc''' or '''c''')<ref name="Fitzpatrick_2021" /> or to one '''revolution''' (symbol '''rev''' or '''r''').<ref name="IET_2016" /> Common related [[Frequency#Unit|units of frequency]] are ''[[cycles per second]]'' (cps) and ''[[revolutions per minute]]'' (rpm).{{efn|The angular unit terms "cycles" and "revolutions" are also used, ambiguously, as shorter versions of the related frequency units.{{cn|date=July 2023}}}} The angular unit of the turn is useful in connection with, among other things, [[electromagnetic coil]]s (e.g., [[transformer]]s), rotating objects, and the [[winding number]] of curves. 
Divisions of a turn include the half-turn and quarter-turn, spanning a [[Angle#Individual angles|straight angle]] and a [[right angle]], respectively; [[metric prefixes]] can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

In the [[International System of Quantities|ISQ]], an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a [[dimensionless quantity]] called '''''rotation''''', defined as the [[ratio]] of a given angle and a full turn. It is represented by the symbol ''N''. {{xref|(See [[#In the ISQ/SI|below]] for the formula.)}}

Because one turn is <math>2\pi</math> radians, some have proposed representing <math>2\pi</math> with the single letter [[Tau (mathematics)|tau]] (<math>\tau</math>).

== Unit symbols ==

There are several unit symbols for the turn.

=== EU and Switzerland ===
The German standard [[DIN 1315]] (March 1974) proposed the unit symbol "pla" (from Latin: {{lang|la|plenus angulus}} 'full angle') for turns.<ref name="German_2013"/><ref name="Kurzweil_1999"/> Covered in {{ill|DIN 1301-1|de}} (October 2010)<!-- in a table "allgemein anwendbare Einheiten außerhalb des SI" -->, the so-called {{lang|de|Vollwinkel}} ('full angle') is not an [[SI unit]]. However, it is a [[legal unit of measurement]] in the EU<ref name="EWG_1980"/><ref name="EG_2009"/> and Switzerland.<ref name="Einheitenverordnung_1994"/>

=== Calculators ===
The scientific calculators [[HP&nbsp;39gII]] and [[HP&nbsp;Prime]] support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to [[newRPL]] for the [[HP&nbsp;50g]] in 2016, and for the [[hp&nbsp;39g+]], [[HP&nbsp;49g+]], [[HP&nbsp;39gs]], and [[HP&nbsp;40gs]] in 2017.<ref name="Lapilli_2016"/><ref name="Lapilli_2018"/> An angular mode <var>TURN</var> was suggested for the [[WP&nbsp;43S]] as well,<ref name="Paul_2016"/> but the calculator instead implements "MUL{{pi}}" (''[[multiples of π|multiples of {{pi}}]]'') as mode and unit since 2019.<ref name="Bonin_2019_OG"/><ref name="Bonin_2019_RG"/>

== Divisions ==
{{see also|Angle#Units}}

Many angle units are defined as a division of the turn. For example, the [[Degree (angle)|degree]] is defined such that one turn is 360 degrees.

Using [[metric prefix]]es, the turn can be divided in 100 centiturns or {{val|1000}} milliturns, with each milliturn corresponding to an [[angle]] of 0.36°, which can also be written as [[Minute and second of arc|21′&nbsp;36″]].<ref name="Hoyle_1962" /><ref name="Klein_2012" /> A [[protractor]] divided in centiturns is normally called a "[[percentage]] protractor". While percentage protractors have existed since 1922,<ref name="Croxton_1992" /> the terms centiturns, milliturns and microturns<!-- ca. 1.3" --> were introduced much later by the British astronomer [[Fred Hoyle]] in 1962.<ref name="Hoyle_1962" /><ref name="Klein_2012" /> Some measurement devices for artillery and [[satellite watching]] carry milliturn scales.<ref name="Schiffner_1965" /><ref name="Hayes_1975" />

[[Binary angular measurement|Binary fractions of a turn]] are also used. Sailors have traditionally divided a turn into 32 [[points of the compass|compass points]], which implicitly have an angular separation of {{sfrac|1|32}}&nbsp;turn. The ''binary degree'', also known as the ''[[binary radian]]'' (or ''brad''), is {{sfrac|1|256}}&nbsp;turn.<ref name="Savage_2007" /> 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 {{math|2<sup>''n''</sup>}} equal parts for other values of {{mvar|n}}.<ref name="Hargreaves_2010" />

== Unit conversion ==
[[File:2pi-unrolled.gif|400px|thumb|right|The [[circumference]] of the [[unit circle]] (whose [[radius]] is one) is {{math|2''π''}}.]]

One turn is equal to <math>2\pi</math> = <math>\tau</math> ≈ {{val|6.283185307179586}}<ref name="OEIS2C_A019692" /> [[radian]]s, 360 [[degree (angle)|degrees]], or 400 [[gradian]]s.

{| class="wikitable" style="text-align:center;"
|+ Conversion of common angles
|-
! Turns
! colspan="2" | [[Radian]]s
! [[Degree (angle)|Degree]]s
! [[Gradian]]s
|-
| 0 turn
| colspan="2" | 0 rad
| 0°
| 0<sup>g</sup>
|-
| {{sfrac|1|72}} turn
| {{sfrac|{{tau}}|72}} rad
| {{sfrac|{{pi}}|36}} rad
| 5°
| {{sfrac|5|5|9}}<sup>g</sup>
|-
| {{sfrac|1|24}} turn
| {{sfrac|{{tau}}|24}} rad
| {{sfrac|{{pi}}|12}} rad
| 15°
| {{sfrac|16|2|3}}<sup>g</sup>
|-
| {{sfrac|1|16}} turn
| {{sfrac|{{tau}}|16}} rad
| {{sfrac|{{pi}}|8}} rad
| 22.5°
| 25<sup>g</sup>
|-
| {{sfrac|1|12}} turn
| {{sfrac|{{tau}}|12}} rad
| {{sfrac|{{pi}}|6}} rad
| 30°
| {{sfrac|33|1|3}}<sup>g</sup>
|-
| {{sfrac|1|10}} turn
| {{sfrac|{{tau}}|10}} rad
| {{sfrac|{{pi}}|5}} rad
| 36°
| 40<sup>g</sup>
|-
| {{sfrac|1|8}} turn
| {{sfrac|{{tau}}|8}} rad
| {{sfrac|{{pi}}|4}} rad
| 45°
| 50<sup>g</sup>
|-
| {{sfrac|1|2{{pi}}}} turn
| colspan="2" | 1 rad
| {{circa}} 57.3°
| {{circa}} 63.7<sup>g</sup>
|-
| {{sfrac|1|6}} turn
| {{sfrac|{{tau}}|6}} rad
| {{sfrac|{{pi}}|3}} rad
| 60°
| {{sfrac|66|2|3}}<sup>g</sup>
|-
| {{sfrac|1|5}} turn
| {{sfrac|{{tau}}|5}} rad
| {{sfrac|2{{pi}}|5}} rad
| 72°
| 80<sup>g</sup>
|-
| {{sfrac|1|4}} turn
| {{sfrac|{{tau}}|4}} rad
| {{sfrac|{{pi}}|2}} rad
| 90°
| 100<sup>g</sup>
|-
| {{sfrac|1|3}} turn
| {{sfrac|{{tau}}|3}} rad
| {{sfrac|2{{pi}}|3}} rad
| 120°
| {{sfrac|133|1|3}}<sup>g</sup>
|-
| {{sfrac|2|5}} turn
| {{sfrac|2{{tau}}|5}} rad
| {{sfrac|4{{pi}}|5}} rad
| 144°
| 160<sup>g</sup>
|-
| {{sfrac|1|2}} turn
| {{sfrac|{{tau}}|2}} rad
| {{pi}} rad
| 180°
| 200<sup>g</sup>
|-
| {{sfrac|3|4}} turn
| {{sfrac|3{{tau}}|4}} rad
| {{sfrac|3{{pi}}|2}} rad
| 270°
| 300<sup>g</sup>
|-
| 1 turn
| {{tau}} rad
| 2{{pi}} rad
| 360°
| 400<sup>g</sup>
|}

== In the ISQ/SI ==
{{anchor|In_the_ISQ/SI}}
{{Infobox physical quantity
| name = Rotation
| othernames = number of revolutions, number of cycles, number of turns, number of rotations
| width =
| background =
| image = 
| caption = 
| unit = [[Unitless]]
| otherunits =
| symbols = ''N''
| baseunits =
| dimension = [[Dimension one|1]]
| extensive =
| intensive =
| conserved =
| transformsas =
| derivations =
}}

In the [[International System of Quantities]] (ISQ), '''rotation''' (symbol '''''N''''') is a [[physical quantity]] defined as '''number of revolutions''':<ref name="ISO80000-3_2019" />

<blockquote>''N'' is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:<!-- Difference from the original formula present in the following formula is intentional. See [https://en.wikipedia.org/wiki/Talk:Turn_(angle)#Way_forward]. -->
: <math>N = \frac{\varphi}{2 \pi \text{ rad}}</math>
where {{varphi}} denotes the measure of [[rotational displacement]].</blockquote>

The above definition is part of the ISQ, formalized in the international standard [[ISO 80000-3]] (Space and time),<ref name="ISO80000-3_2019" /> and adopted in the [[International System of Units]] (SI).<ref name="SIBrochure_9" /><ref name="NISTGuide_2009" />

Rotation count or number of revolutions is a [[quantity of dimension one]], resulting from a ratio of angular displacement.
It can be negative and also greater than 1 in modulus.
The relationship between quantity rotation, ''N'', and unit turns, tr, can be expressed as:
: <math>N = \frac \varphi \text{tr} = \{ \varphi \}_\text{tr}</math>
where <nowiki>{</nowiki>{{varphi}}<nowiki>}</nowiki><sub>tr</sub> is the numerical value of the angle {{varphi}} in units of turns (see ''{{slink|Physical quantity#Components}}'').

In the ISQ/SI, rotation is used to derive [[rotational frequency]] (the [[Derivative|rate of change]] of rotation with respect to time), denoted by {{mvar|n}}:
: <math>n = \frac{\mathrm{d}N}{\mathrm{d}t}</math>

The SI unit of rotational frequency is the [[reciprocal second]] (s<sup>−1</sup>). Common related [[Frequency#Unit|units of frequency]] are ''[[hertz]]'' (Hz), ''[[cycles per second]]'' (cps), and ''[[revolutions per minute]]'' (rpm).

{{Infobox unit
| name          = Revolution
| othernames    = Cycle
| standard      = 
| quantity      = [[Rotation (quantity)|Rotation]]
| symbol        = rev
| symbol2       = r
| symbol3       = cyc
| symbol4       = c
| units1        = [[Base unit (measurement)|Base unit]]s
| inunits1      = [[One (unit)|1]]
}}

{{anchor|Rotational unit}}
The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the [[dimensionless unit]] "one",{{efn|"The special name revolution, symbol r, for this unit [name 'one', symbol '1'] is widely used in specifications on rotating machines."<ref name="ISO 80000-3:2006">{{cite web | title=ISO 80000-3:2006 | website=ISO | date=2001-08-31 | url=https://www.iso.org/standard/31888.html | access-date=2023-04-25}}</ref>}}
which also received other special names, such as the radian.{{efn|"Measurement units of quantities of dimension one are numbers. In some cases, these measurement units are given special names, e.g. radian..."<ref name="ISO 80000-3:2006"/>}}
Despite their [[dimensional homogeneity]], these two specially named dimensionless units are applicable for non-comparable [[kind of quantity|kinds of quantity]]: rotation and angle, respectively.<ref name="ISO 80000-1">{{cite web |title=ISO 80000-1:2009(en) Quantities and units — Part 1: General |url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-1:v1:en |access-date=2023-05-12 |website=iso.org}}</ref>
"Cycle" is also mentioned in ISO 80000-3, in the definition of ''[[period (physics)|period]]''.{{efn|"3-14) period duration, period: duration (item 3‑9) of one cycle of a periodic event"<ref name="ISO80000-3_2019"/>}}

== See also ==
* [[Ampere-turn]]
* [[Hertz]] (modern) or [[Cycle per second]] (older)
* [[Angle of rotation]]
* [[Revolutions per minute]]
* [[Repeating circle]]
* [[Spat (angular unit)]] – the [[solid angle]] counterpart of the turn, equivalent to {{math|4''π''}}&nbsp;[[steradian]]s.
* [[Unit interval]]
* ''[[Divine Proportions: Rational Trigonometry to Universal Geometry]]''
* [[Modulo operation]]
* [[Twist (rational trigonometry)|Twist (mathematics)]]
* [[Tau (mathematics)]]

== Notes ==
{{notelist}}

== References ==
{{reflist|refs=

<ref name="Savage_2007">{{cite web |title=ooPIC Programmer's Guide - Chapter 15: URCP |work=[[ooPIC]] Manual & Technical Specifications - ooPIC Compiler Ver 6.0 |orig-date=1997 |date=2007 |publisher=Savage Innovations, LLC |url=http://www.oopic.com/pgchap15.htm |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm |archive-date=2008-06-28}}</ref>

<ref name="Hargreaves_2010">{{cite web |title=Angles, integers, and modulo arithmetic |author-first=Shawn |author-last=Hargreaves |author-link=:pl:Shawn Hargreaves |publisher=blogs.msdn.com |url=http://blogs.msdn.com/shawnhar/archive/2010/01/04/angles-integers-and-modulo-arithmetic.aspx |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190630223817/http://www.shawnhargreaves.com/blogindex.html |archive-date=2019-06-30}}</ref>

<ref name="Croxton_1992">{{cite journal |author-first=Frederick E. |author-last=Croxton |date=1922 |title=A Percentage Protractor - Designed for Use in the Construction of Circle Charts or "Pie Diagrams" |series=Short Note |journal=[[Journal of the American Statistical Association]] |volume=18 |issue=137 |pages=108–109 |doi=10.1080/01621459.1922.10502455}}</ref>

<ref name="Hoyle_1962">{{cite book |author-first=Fred |author-last=Hoyle |author-link=Fred Hoyle |editor-first=M. H. |editor-last=Chandler |title=Astronomy |url=https://archive.org/details/astronom00hoyl |url-access=registration |publisher=[[Macdonald & Co. (Publishers) Ltd.]] / Rathbone Books Limited |location=London, UK |date=1962 |edition=1 |lccn=62065943 |oclc=7419446}} (320 pages)</ref>

<ref name="Klein_2012">{{cite book |author-first=Herbert Arthur |author-last=Klein |title=The Science of Measurement: A Historical Survey (The World of Measurements: Masterpieces, Mysteries and Muddles of Metrology) |chapter=Chapter 8: Keeping Track of Time |edition=corrected reprint of original |date=2012 |orig-date=1988, 1974 |lccn=88-25858 |publisher=[[Dover Publications, Inc.]] / [[Courier Corporation]] (originally by [[Simon & Schuster, Inc.]]) |series=Dover Books on Mathematics |isbn=978-0-48614497-9 |page=102 |chapter-url=https://books.google.com/books?id=CrmuSiCFyikC&pg=PA102 |access-date=2019-08-06}} (736 pages)</ref>

<ref name="Schiffner_1965">{{cite journal |title=Bestimmung von Satellitenbahnen |language=de |author-first=Friedrich |author-last=Schiffner |editor-first=Maria Emma |editor-last=Wähnl |editor-link=:de:Maria Emma Wähnl |journal=[[Astronomische Mitteilungen der Urania-Sternwarte Wien]] |publisher=[[Volksbildungshaus Wiener Urania]] |location=Wien, Austria |volume=8 |issue= |date=1965}}</ref>

<ref name="Hayes_1975">{{cite book |title=Trackers of the Skies |author-first=Eugene Nelson |author-last=Hayes |series=History of the Smithsonian Satellite-tracking Program |publisher=[[Academic Press]] / Howard A. Doyle Publishing Company |location=Cambridge, Massachusetts, USA |date=1975 |orig-date=1968 |url=https://siris-sihistory.si.edu/ipac20/ipac.jsp?&profile=all&source=~!sichronology&uri=full=3100001~!3190~!0#focus}}</ref>

<ref name="EWG_1980">{{cite web |title=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 |language=de |date=1980-02-15 |url=https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:01980L0181-20090527 |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190622210052/https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:01980L0181-20090527 |archive-date=2019-06-22}}</ref>

<ref name="EG_2009">{{cite web |title=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) |language=de |date=2009-03-11 |url=https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:32009L0003&from=EN |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190806184426/https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:32009L0003&from=EN |archive-date=2019-08-06}}</ref>

<ref name="Einheitenverordnung_1994">{{cite book |title=Einheitenverordnung |chapter=Art. 15 Einheiten in Form von nichtdezimalen Vielfachen oder Teilen von SI-Einheiten |id=941.202 |date=1994-11-23 |language=de-ch |publisher=[[Schweizerischer Bundesrat]]<!-- |work=Der Bundesrat - Das Portal der Schweizer Regierung--> |chapter-url=http://www.admin.ch/opc/de/classified-compilation/19940345/ |access-date=2013-01-01 |url-status=live |archive-url=https://web.archive.org/web/20190510122902/https://www.admin.ch/opc/de/classified-compilation/19940345/ |archive-date=2019-05-10}}</ref>

<ref name="German_2013">{{cite book |title=Handbuch SI-Einheiten: Definition, Realisierung, Bewahrung und Weitergabe der SI-Einheiten, Grundlagen der Präzisionsmeßtechnik |author-first1=Sigmar |author-last1=German |author-first2=Peter |author-last2=Drath |publisher=[[Friedrich Vieweg & Sohn Verlagsgesellschaft mbH]], reprint: [[Springer-Verlag]] |language=de |date=2013-03-13 |orig-date=1979 |edition=1 |isbn=978-3-32283606-9 |id=978-3-528-08441-7, 978-3-32283606-9 |page=421 |url=https://books.google.com/books?id=63qcBgAAQBAJ&pg=PA421 |access-date=2015-08-14}}</ref>

<ref name="Kurzweil_1999">{{cite book |title=Das Vieweg Einheiten-Lexikon: Formeln und Begriffe aus Physik, Chemie und Technik |author-first=Peter |author-last=Kurzweil |language=de |publisher=Vieweg, reprint: [[Springer-Verlag]] |edition=1 |date=2013-03-09 |orig-date=1999 |isbn=978-3-32292920-4 |id=978-3-322-92921-1 |doi=10.1007/978-3-322-92920-4 |page=403 |url=https://books.google.com/books?id=2zecBgAAQBAJ |access-date=2015-08-14}}</ref>

<ref name="Lapilli_2016">{{cite web |title=RE: newRPL: Handling of units |author-first=Claudio Daniel |author-last=Lapilli |date=2016-05-11 |work=HP Museum |url=http://www.hpmuseum.org/forum/thread-4783-post-55836.html#pid55836 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20170810012742/http://www.hpmuseum.org/forum/thread-4783-post-55836.html |archive-date=2017-08-10}}</ref>

<ref name="Lapilli_2018">{{cite book |title=newRPL User Manual |chapter=Chapter 3: Units - Available Units - Angles |author-first=Claudio Daniel |author-last=Lapilli |date=2018-10-25 |chapter-url=https://newrpl.wiki.hpgcc3.org/doku.php?id=manual:chapter3:units#available-units |access-date=2019-08-07 |url-status=live |archive-url=https://web.archive.org/web/20190806225910/https://newrpl.wiki.hpgcc3.org/doku.php?id=manual:chapter3:units#available-units |archive-date=2019-08-06}}</ref>

<ref name="OEIS2C_A019692">Sequence {{OEIS2C|A019692}}</ref>

<ref name="Paul_2016">{{cite web |title=RE: WP-32S in 2016? |date=2016-01-12 |orig-date=2016-01-11 |author-first=Matthias R. |author-last=Paul |work=HP Museum |url=https://www.hpmuseum.org/forum/thread-5427-post-48945.html#pid48945 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190805163709/https://www.hpmuseum.org/forum/thread-5427-post-48945.html |archive-date=2019-08-05 |quote=[…] I'd like to see a TURN mode being implemented as well. TURN mode works exactly like DEG, RAD and GRAD (including having a full set of angle unit conversion functions like on the [[WP 34S]]), except for that a full circle doesn't equal 360 degree, 6.2831... rad or 400 gon, but 1 turn. (I […] found it to be really convenient in engineering/programming, where you often have to convert to/from other unit representations […] But I think it can also be useful for educational purposes. […]) Having the angle of a full circle normalized to 1 allows for easier [[Rule of three (mathematics)|conversions]] to/from a whole bunch of other angle units […]}}</ref>

<ref name="Bonin_2019_OG">{{cite book |title=WP&nbsp;43S Owner's Manual |date=2019 |orig-date=2015 |author-last=Bonin |author-first=Walter |isbn=978-1-72950098-9 |edition=draft |version=0.12 |pages=72, 118–119, 311 |url=https://gitlab.com/wpcalculators/wp43/-/raw/master/docs/OwnersManual.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230718192232/https://gitlab.com/rpncalculators/wp43/-/raw/master/docs/OwnersManual.pdf |archive-date=2023-07-18}} [https://gitlab.com/Over_score/wp43s] [https://gitlab.com/wpcalculators/wp43] (314 pages)</ref>

<ref name="Bonin_2019_RG">{{cite book |title=WP&nbsp;43S Reference Manual |date=2019 |orig-date=2015 |author-last=Bonin |author-first=Walter |isbn=978-1-72950106-1 |edition=draft |version=0.12 |pages=iii, 54, 97, 128, 144, 193, 195 |url=https://gitlab.com/wpcalculators/wp43/-/raw/master/docs/ReferenceManual.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230718192332/https://gitlab.com/rpncalculators/wp43/-/raw/master/docs/ReferenceManual.pdf |archive-date=2023-07-18}} [https://gitlab.com/Over_score/wp43s] [https://gitlab.com/wpcalculators/wp43] (271 pages)</ref>

<ref name="Fitzpatrick_2021">{{cite book |author-last=Fitzpatrick |author-first=Richard |title=Newtonian Dynamics: An Introduction |publisher=[[CRC Press]] |date=2021 |isbn=978-1-000-50953-3 |url=https://books.google.com/books?id=rRpSEAAAQBAJ&pg=PA116 |access-date=2023-04-25 |page=116}}</ref>

<ref name="IET_2016">{{cite book |title=Units & Symbols for Electrical & Electronic Engineers |date=2016 |publisher=[[Institution of Engineering and Technology]] |publication-place=London, UK |url=https://www.theiet.org/media/4173/units-and-symbols.pdf |access-date=2023-07-18 |url-status=live |archive-url=https://web.archive.org/web/20230718183635/https://www.theiet.org/media/4173/units-and-symbols.pdf |archive-date=2023-07-18}} (1+iii+32+1 pages)</ref>

<ref name="ISO80000-3_2019">{{cite web |title=ISO 80000-3:2019 Quantities and units — Part 3: Space and time |publisher=[[International Organization for Standardization]] |date=2019 |edition=2 |url=https://www.iso.org/standard/64974.html |access-date=2019-10-23}} [https://www.iso.org/obp/ui/#iso:std:iso:80000:-3:ed-2:v1:en] (11 pages)</ref>

<ref name="SIBrochure_9">{{SIbrochure9th}}</ref>

<ref name="NISTGuide_2009">{{cite web |title=The NIST Guide for the Use of the International System of Units, Special Publication 811 |author-first1=Ambler |author-last1=Thompson |author-first2=Barry N. |author-last2=Taylor |edition=2008 |publisher=[[National Institute of Standards and Technology]] |date=2020-03-04 |orig-date=2009-07-02 |ref={{sfnref|NIST|2009}} |url=https://www.nist.gov/pml/special-publication-811 |access-date=2023-07-17}} [https://web.archive.org/web/20230515201622/https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication811e2008.pdf]</ref>

}} <!-- END reflist -->

== External links ==
* [https://tauday.com/tau-manifesto ''The Tau Manifesto'']

{{DEFAULTSORT:Turn (Geometry)}}
[[Category:Units of plane angle]]
[[Category:Mathematical concepts]]
[[Category:Angle]]
[[Category:1 (number)]]