| Turn | |
|---|---|
 Counterclockwiserotations about the center point starting from the right, where a complete rotation corresponds to an angle of rotation of 1 turn. | |
| General information | |
| Unit of | Plane angle |
| Symbol | tr, pla, rev, cyc |
| Conversions | |
| 1 trin ... | ... is equal to ... |
| radians | 2π rad ≈6.283185307... rad |
| milliradians | 2000π mrad ≈6283.185307... mrad |
| degrees | 360° |
| gradians | 400g |
Theturn (symboltr orpla) is a unit ofplane angle measurement that is the measure of acomplete angle—the anglesubtended by a completecircle at its center. One turn is equal to2π radians, 360 degrees or 400 gradians. As anangular unit, one turn also corresponds to onecycle (symbolcyc orc)[1] or to onerevolution (symbolrev orr).[2] Common relatedunits of frequency arecycles per second (cps) andrevolutions per minute (rpm). The angular unit of the turn is useful in connection with, among other things,electromagnetic coils (e.g.,transformers), rotating objects, and thewinding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning astraight angle and aright angle, respectively;metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.
In theISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as adimensionless quantity calledrotation, defined as theratio of a given angle and a full turn. It is represented by the symbolN.(Seebelow for the formula.)
Because one turn is radians, some have proposed representing with the single letter𝜏 (tau).[3]
There are several unit symbols for the turn.
The German standardDIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin:plenus angulus 'full angle') for turns.[4][5] Covered inDIN 1301-1 [de] (October 2010), the so-calledVollwinkel ('full angle') is not anSI unit. However, it is alegal unit of measurement in the EU[6][7] and Switzerland.[8]
The scientific calculatorsHP 39gII andHP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added tonewRPL for theHP 50g in 2016, and for thehp 39g+,HP 49g+,HP 39gs, andHP 40gs in 2017.[9][10] An angular modeTURN was suggested for theWP 43S as well,[11] but the calculator instead implements "MULπ" (multiples ofπ) as mode and unit since 2019.[12][13]
Many angle units are defined as a division of the turn. For example, thedegree is defined such that one turn is 360 degrees.
Usingmetric prefixes, the turn can be divided in 100 centiturns or1000 milliturns, with each milliturn corresponding to anangle of 0.36°, which can also be written as21′ 36″.[14][15] Aprotractor divided in centiturns is normally called a "percentage protractor". While percentage protractors have existed since 1922,[16] the terms centiturns, milliturns and microturns were introduced much later by the British astronomerFred Hoyle in 1962.[14][15] Some measurement devices for artillery andsatellite watching carry milliturn scales.[17][18]
Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32compass points, which implicitly have an angular separation of1/32 turn. Thebinary degree, also known as thebinary radian (orbrad), is1/256 turn.[19] The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a singlebyte. Other measures of angle used in computing may be based on dividing one whole turn into2n equal parts for other values ofn.[20]
One turn is equal to = ≈6.283185307179586[21]radians, 360degrees, or 400gradians.
| Turns | Radians | Degrees | Gradians | |
|---|---|---|---|---|
| 0 turn | 0 rad | 0° | 0g | |
| 1/72 turn | 𝜏/72 rad | π/36 rad | 5° | 5+5/9g |
| 1/24 turn | 𝜏/24 rad | π/12 rad | 15° | 16+2/3g |
| 1/16 turn | 𝜏/16 rad | π/8 rad | 22.5° | 25g |
| 1/12 turn | 𝜏/12 rad | π/6 rad | 30° | 33+1/3g |
| 1/10 turn | 𝜏/10 rad | π/5 rad | 36° | 40g |
| 1/8 turn | 𝜏/8 rad | π/4 rad | 45° | 50g |
| 1/2π turn | 1 rad | c. 57.3° | c. 63.7g | |
| 1/6 turn | 𝜏/6 rad | π/3 rad | 60° | 66+2/3g |
| 1/5 turn | 𝜏/5 rad | 2π/5 rad | 72° | 80g |
| 1/4 turn | 𝜏/4 rad | π/2 rad | 90° | 100g |
| 1/3 turn | 𝜏/3 rad | 2π/3 rad | 120° | 133+1/3g |
| 2/5 turn | 2𝜏/5 rad | 4π/5 rad | 144° | 160g |
| 1/2 turn | 𝜏/2 rad | π rad | 180° | 200g |
| 3/4 turn | 3𝜏/4 rad | 3π/2 rad | 270° | 300g |
| 1 turn | 𝜏 rad | 2π rad | 360° | 400g |
| Rotation | |
|---|---|
Other names | number of revolutions, number of cycles, number of turns, number of rotations |
Common symbols | N |
| SI unit | Unitless |
| Dimension | 1 |
In theInternational System of Quantities (ISQ),rotation (symbolN) is aphysical quantity defined asnumber of revolutions:[22]
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:
where 𝜑 denotes the measure ofrotational displacement.
The above definition is part of the ISQ, formalized in the international standardISO 80000-3 (Space and time),[22] and adopted in theInternational System of Units (SI).[23][24]
Rotation count or number of revolutions is aquantity 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:
where {𝜑}tr is the numerical value of the angle 𝜑 in units of turns (seePhysical quantity § Components).
In the ISQ/SI, rotation is used to deriverotational frequency (therate of change of rotation with respect to time), denoted byn:
The SI unit of rotational frequency is thereciprocal second (s−1). Common relatedunits of frequency arehertz (Hz),cycles per second (cps), andrevolutions per minute (rpm).
| Revolution | |
|---|---|
| Unit of | Rotation |
| Symbol | rev, r, cyc, c |
| Conversions | |
| 1 revin ... | ... is equal to ... |
| Base units | 1 |
The superseded version ISO 80000-3:2006 defined "revolution" as a special name for thedimensionless unit "one",[a]which also received other special names, such as the radian.[b]Despite theirdimensional homogeneity, these two specially named dimensionless units are applicable for non-comparablekinds of quantity: rotation and angle, respectively.[26]"Cycle" is also mentioned in ISO 80000-3, in the definition ofperiod.[c]
[…] 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 theWP 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 easierconversions to/from a whole bunch of other angle units […]