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Turn (angle)
Turn (angle)
Turn (angle)
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Turn
Counterclockwise rotations about the center point starting from the right, where a complete rotation corresponds to an angle of rotation of 1 turn.
General information
Unit ofPlane angle
Symboltr, pla, rev, cyc
Conversions
1 tr in ...... is equal to ...
   radians   2π rad
≈ 6.283185307... rad
   milliradians   2000π mrad
≈ 6283.185307... mrad
   degrees   360°
   gradians   400g

The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to 2π radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c)[1] or to one revolution (symbol rev or r).[2] Common related units of frequency are cycles per second (cps) and revolutions 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 the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

In the 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. (See below for the formula.)

Because one turn is radians, some have proposed representing with the single letter 𝜏 (tau).[3]

Unit symbols

[edit]

There are several unit symbols for the turn.

EU and Switzerland

[edit]

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: plenus angulus 'full angle') for turns.[4][5] Covered in DIN 1301-1 [de] (October 2010), the so-called Vollwinkel ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU[6][7] and Switzerland.[8]

Calculators

[edit]

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.[9][10] An angular mode TURN was suggested for the WP 43S as well,[11] but the calculator instead implements "MULπ" (multiples of π) as mode and unit since 2019.[12][13]

Divisions

[edit]
See also: Angle § Units

Many angle units are defined as a division of the turn. For example, the degree is defined such that one turn is 360 degrees.

Using metric prefixes, the 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″.[14][15] A protractor 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 astronomer Fred Hoyle in 1962.[14][15] Some measurement devices for artillery and satellite watching carry milliturn scales.[17][18]

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of ⁠1/32⁠ turn. The binary degree, also known as the binary radian (or brad), is ⁠1/256⁠ turn.[19] 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.[20]

Unit conversion

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

One turn is equal to = ≈ 6.283185307179586[21] radians, 360 degrees, or 400 gradians.

Conversion of common angles
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

In the ISQ/SI

[edit]

Rotation
Other names
number of revolutions, number of cycles, number of turns, number of rotations
Common symbols
N
SI unitUnitless
Dimension1

In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number 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 of rotational displacement.

The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time),[22] and adopted in the International System of Units (SI).[23][24]

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:

where {𝜑}tr is the numerical value of the angle 𝜑 in units of turns (see Physical quantity § Components).

In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by n:

The SI unit of rotational frequency is the reciprocal second (s−1). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

Revolution
Unit ofRotation
Symbolrev, r, cyc, c
Conversions
1 rev in ...... is equal to ...
   Base units   1

The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one",[a] which also received other special names, such as the radian.[b] Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.[26] "Cycle" is also mentioned in ISO 80000-3, in the definition of period.[c]

See also

[edit]

Notes

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Turn (angle)

View on Grokipedia
from Grokipedia
A turn is a unit of plane angle measurement corresponding to one complete rotation of a ray or line segment around a fixed point, returning it to its original position. This unit, also known as a revolution, full turn, or cycle, represents the full circumference of a circle in angular terms. In standard angular systems, one turn equals 360 degrees, 2π radians, or 400 gradians (also called gons). These equivalences arise from historical and mathematical conventions: the degree system divides the turn into 360 equal parts, the radian measures the arc length equal to the radius (yielding 2π for a full circle), and the gradian divides it into 400 parts for decimal compatibility. Although not an official SI unit—the radian holds that status—the turn is recognized in educational and geometric contexts for its intuitive representation of a complete cycle. The turn serves as a foundational reference in geometry, trigonometry, and rotational dynamics, where fractions like a quarter turn (90 degrees or π/2 radians, a right angle) or half turn (180 degrees or π radians, a straight angle) describe common orientations. Its use simplifies discussions of periodicity in mathematics and science, such as in circular motion or modular arithmetic, by aligning directly with one full cycle without fractional multipliers.

Definition and Notation

Core Definition

A turn is a unit of plane angle measurement defined as the central angle subtended by the complete circumference of a circle, corresponding to one full rotation about the center. This unit captures the intuitive concept of a complete cycle, where a point returns to its initial position after traversing the entire boundary of the circle. In standard angular measures, one turn equals 360 degrees or 2π2\pi radians, but its foundational appeal lies in representing wholeness rather than arbitrary subdivisions of a circle. The etymology of "turn" traces to the late Old English verb turnian, meaning "to rotate" or "revolve," evoking the notion of a full circuit or change in direction through rotation. This planar focus underscores its role in pure geometry, where it serves as a natural benchmark for complete rotations, equivalent to 360 degrees for comparative purposes.

Symbols and Abbreviations

The primary symbol for the turn unit of plane angle is "tr", as designated in the International Standard ISO 80000-3 for quantities and units related to space and time. This symbol facilitates precise notation in scientific and engineering contexts where a full rotation is measured. In certain mathematical discussions, alternative notations include "t" for turn or the Greek letter tau (τ), with tau explicitly defined as the constant 2π radians corresponding to one complete turn around a circle. The use of tau emphasizes the periodicity of circular functions, as proposed in influential works advocating its adoption over traditional constants. Abbreviations vary by context: the unabbreviated word "turn" appears in descriptive prose, whereas the compact "tr" is standard in technical documentation and formulas to denote the unit explicitly. International standards, including ISO 80000-1, specify that unit symbols such as "tr" must be rendered in upright (roman) typeface to distinguish them from italicized symbols for physical quantities or variables. This typographic convention ensures clarity in printed and digital publications. An example of its application in equations is the representation of a full rotation as θ=1 tr\theta = 1\ \mathrm{tr}, where θ\theta denotes the plane angle in turns.

Historical Context

Origins in Geometry and Navigation

The conceptual foundation of the turn as a complete angular rotation emerged in ancient geometry through the work of Babylonian mathematicians. The Babylonians, employing a sexagesimal (base-60) number system, divided the full circumference of a circle into 360 equal parts for astronomical observations of the ecliptic, implicitly defining the entire circle as one unified rotation or turn. This division facilitated precise tracking of celestial movements and was highly divisible, allowing for subdivisions like 60 arcminutes per part, which influenced subsequent geometric practices. The Greeks later adopted and formalized this 360-division in geometry, as seen in the works of Euclid and Ptolemy, applying it to astronomy and early navigation. In maritime navigation during the 16th and 17th centuries, navigation fundamentally relies on angular measurement derived from rotational turns, as adjusting a ship's rudder by a specific angle induces a corresponding turn in the vessel's heading, with the full circle representing a complete 360-degree rotation back to the original bearing. This usage aligned with compass roses divided into 32 points encompassing the full turn, aiding in dead reckoning and course plotting during explorations. By the 18th century, clockmaking further reinforced the standardization of the full circle as one turn through mechanical and tabular representations. Clock mechanisms, such as those developed with pendulum regulators, required calibrating gear ratios for the hour hand to complete two full turns daily, embedding the concept of a complete rotation in timekeeping devices.

Evolution in Measurement Standards

During the French Revolution in the 1790s, reformers sought to decimalize all measurements as part of the broader metric system overhaul, including angles. A centesimal system was proposed, dividing the full circle—or "turn"—into 400 gradians (also called grades or gons), with a right angle equaling 100 gradians; this aimed to align angular measurement with the decimal base of the new system. However, the traditional sexagesimal degree (360 per full turn) persisted due to its entrenched use in astronomy, navigation, and surveying, rendering the gradian and explicit turn-based divisions supplementary at best. International standardization efforts accelerated in the 1960s with the ISO 31 series on quantities and units, which laid groundwork for modern angle measurement by defining coherent systems. This evolved into the ISO 80000 series. A significant 20th-century advancement came in 2001 when mathematician Bob Palais advocated for the turn in his essay "Pi Is Wrong!", proposing it as the basis for a circle constant τ (tau) equal to one full turn (approximately 6.2832), to streamline radian-based formulas and highlight the natural periodicity of circles over π. This tau movement gained traction in mathematical education, influencing pedagogy by encouraging expressions like "one-quarter turn" for 90 degrees instead of π/2 radians. Key standardization milestones included the ISO 80000-1:2009, which formally recognized the turn (symbol: tr) as a unit for plane angle equivalent to one complete rotation (2π radians), alongside the revolution (symbol: r) for rotating systems. This integrated the turn into the International System of Quantities (ISQ), affirming its status as a non-SI unit acceptable for use with SI, denoted as "tr" to denote one full angular cycle.

Relations to Other Units

Equivalence to Degrees and Radians

The turn is a unit of angular measure equivalent to one complete rotation, corresponding exactly to 360 degrees, 2π radians, and 400 gradians (also known as gons). This equivalence derives from the geometric definition of angular measure based on the circle's circumference. The angle in turns is given by the ratio of the arc length ss to the full circumference 2πr2\pi r, where rr is the radius: θturns=s2πr.\theta_{\text{turns}} = \frac{s}{2\pi r}. For a full rotation, s=2πrs = 2\pi r, yielding θ=1\theta = 1 turn, establishing the turn as the natural unit for a complete circle. For partial angles, the relationships simplify conversions as follows: θturns=θdegrees360=θradians2π.\theta_{\text{turns}} = \frac{\theta_{\text{degrees}}}{360} = \frac{\theta_{\text{radians}}}{2\pi}. These formulas allow direct scaling from other units to fractions of a turn. Using turns offers advantages in calculations involving periodic phenomena, where full cycles align with integer values, facilitating analysis of periodicity without fractional adjustments in radians or degrees.

Subdivisions and Fractions

The turn, as a unit of angular measure, is often subdivided into binary fractions that align with fundamental geometric concepts. The half-turn, equivalent to 180° or π radians, represents a complete reversal or straight angle, commonly applied in geometry for describing reflections and 180-degree rotations in shapes and transformations. Similarly, the quarter-turn, corresponding to 90° or π/2 radians, denotes a right angle and is essential for constructing perpendicular lines, squares, and coordinate axes in Euclidean geometry. These binary fractions provide intuitive divisions, as powers of two facilitate halving and quartering in visual and diagrammatic representations without requiring decimal approximations. For finer precision, the centiturn (ct), defined as 1/100 of a turn or 3.6°, serves as a decimal subdivision proposed by astronomer Fred Hoyle in 1962 to enhance accuracy in observational measurements. This unit, also known as a percentage of a full circle, is employed in specialized protractors and instrumentation, offering a metric-like granularity suitable for applications demanding sub-degree resolution, such as alignment in technical drawings. Beyond binary and decimal divisions, other fractions like the third-turn, equal to 120° or 2π/3 radians, arise in contexts of rotational symmetry and periodic functions. In symmetry, the third-turn characterizes the order-3 rotational invariance of equilateral triangles, where a 120-degree rotation maps the figure onto itself, a property central to group theory and crystallographic analysis. Trigonometric identities, such as those for 120-degree angles in the unit circle, further utilize this fraction to express roots of unity and waveform repetitions. In computing, angles measured in turns benefit from alignment with modular arithmetic, where operations modulo 1 naturally handle periodic wrapping without scaling factors, simplifying implementations in algorithms for graphics rendering, animation, and numerical simulations. Binary fractions of a turn, such as 1/256, integrate seamlessly with digital representations, reducing computational overhead in fixed-point arithmetic for angular computations.

Conversions and Calculations

Unit Conversion Formulas

The primary conversion formulas between turns (or revolutions, symbol r) and other common angular units derive from the definition of one turn as a complete rotation. Specifically, 1 turn = 360° and 1 turn = 2π rad. To convert from degrees to turns, divide the angle in degrees by 360: turns=degrees360\text{turns} = \frac{\text{degrees}}{360} Conversely, to convert from turns to degrees, multiply the number of turns by 360: degrees=turns×360\text{degrees} = \text{turns} \times 360 For conversions involving radians, divide the angle in radians by 2π2\pi to obtain turns: turns=radians2π\text{turns} = \frac{\text{radians}}{2\pi} To convert turns to radians, multiply by 2π2\pi: radians=turns×2π\text{radians} = \text{turns} \times 2\pi The gradian (also known as gon), a unit where 400 gradians comprise one full turn, allows for direct conversion by dividing gradians by 400: turns=gradians400\text{turns} = \frac{\text{gradians}}{400} This relation stems from the gradian's definition as π/200\pi/200 radians, aligning with the turn's equivalence to 2π2\pi radians. Multi-step conversions can be performed using an intermediate unit like turns for efficiency, particularly when bridging disparate systems such as gradians to radians. For example, first convert gradians to turns as above, then multiply by 2π2\pi to reach radians. Such chaining minimizes intermediate approximations but requires consistent precision across steps. Numerical conversions involving radians and turns are sensitive to the approximation of π\pi, as the factor 2π2\pi is irrational. Using a low-precision value like π≈3.1416\pi \approx 3.1416 (absolute error of about 0.000007) in a conversion from 1 radian to turns yields 1/(2×3.1416)≈0.1591151 / (2 \times 3.1416) \approx 0.159115 tr, whereas the exact value is 1/(2π)≈0.1591549431/(2\pi) \approx 0.159154943 tr—a relative error of roughly 0.025%. Higher-precision approximations, such as π≈3.1415926535\pi \approx 3.1415926535, reduce this to negligible levels for most applications. As an illustrative example, consider converting 45° to turns. Apply the degrees-to-turns formula: turns=45360\text{turns} = \frac{45}{360} First, simplify the fraction: 45÷45=145 \div 45 = 1, 360÷45=8360 \div 45 = 8, so 45/360=1/845/360 = 1/8. Then, 1/8=0.1251/8 = 0.125. Thus, 45° = 0.125 tr exactly, since 360 divides evenly in this case.

Practical Computation Tools

Various online calculators and converters facilitate efficient computation of turn-based angular measurements by supporting conversions between turns (or revolutions) and other units such as degrees, radians, and gradians. For instance, the CalculatorSoup angle conversion tool allows users to input values in revolutions (equivalent to turns) and convert them to multiple angular units, ensuring compliance with standard measurement practices. Similarly, the Gordon England angle unit calculator explicitly includes "full circle or turn" as an option, enabling precise transformations for engineering and scientific applications. These web-based resources are particularly useful for quick verifications without requiring installed software. In programming environments, software libraries provide robust support for turn-related calculations, often leveraging the constant tau (τ = 2π, representing one full turn) for angular periodicity and trigonometric operations. Python's standard math module, since version 3.6, includes the tau constant for direct use in angle computations, such as normalizing angles modulo one turn via expressions like angle % math.tau. Additionally, the 'angles' library on PyPI offers classes for representing angles in various units, including support for full rotations, which simplifies handling of turn-based periodicity in scientific computing tasks. In MATLAB, built-in functions like wrapTo2Pi handle angular wrapping equivalent to one turn (τ radians), facilitating periodicity in simulations and data analysis, while the Mapping Toolbox supports angle representations convertible to turns through radian-based operations. Mobile applications and spreadsheet tools further democratize turn computations for on-the-go or routine use. Apps like the Android Angle Converter support conversions including "circle" (synonymous with turn or revolution), allowing users to switch between angular units via intuitive interfaces for fieldwork in surveying or mechanics. In Microsoft Excel, while the built-in CONVERT function handles common angle units like degrees and radians, turns can be computed manually using formulas such as =A1/360 to convert degrees to fractional turns, integrating seamlessly into larger engineering spreadsheets. These tools implement standard unit conversion formulas for angles as their foundation. Historically, 20th-century slide rules served as analog computation devices adapted for angular measurements in engineering, featuring logarithmic scales for trigonometry and full-circle calculations. Engineering models, such as those following the Rietz system developed around 1902, included sine and tangent scales calibrated for degrees but readily scalable to turns by aligning with 360-degree or 400-gradian markers, aiding navigation and mechanical design computations before digital tools emerged. The Oughtred Society's documentation highlights how these devices, popularized in the mid-19th century, enabled rapid angular resolutions in fields like artillery and astronomy.

Standardization and Usage

Role in the International System of Quantities (ISQ)

The turn is not an official unit in the International System of Units (SI), where the radian (rad) is the coherent unit for plane angle, treated as dimensionless. However, the turn is recognized in some national and regional standards for convenience in describing complete rotations.

Regional and International Variations

The turn, known as Vollwinkel in German, is defined in the German standard DIN 1301-1:2010 with the symbol "tr" (upright roman typography), equivalent to 2π radians or 360 degrees. It is a legal unit of measurement in the European Union and Switzerland, though not part of the SI. Adoption varies internationally. In the United States and mathematical communities, the Greek letter tau (τ ≈ 6.2832) is advocated as a constant for one full turn (2π radians), promoting its use in education to simplify circular measurements, as proposed in works like the Tau Manifesto. This contrasts with engineering preferences in Europe for "tr". Commonwealth countries generally favor degrees or radians in standards from bodies like the British Standards Institution. The symbol "tr" is supported in scientific calculators such as the HP Prime (since 2013) and in web technologies like CSS, where "turn" is an angular unit (1 turn = 360°).

Applications in Modern Fields

In physics, the turn unit is employed to describe full rotations in electromagnetism, particularly in the design of solenoids where the number of turns represents the complete windings of wire around the core, directly influencing the magnetic field strength via the formula B=μ0nIB = \mu_0 n I, with nn as turns per unit length. This approach simplifies quantification of coil density, as each turn corresponds to a 360° loop that contributes to the axial field inside the solenoid. In quantum mechanics, the concept of turns is crucial for understanding spin-1/2 particles, such as electrons, where a single turn (360° rotation) alters the wavefunction's phase by a sign change, necessitating two full turns (720°) to restore the original state, a phenomenon illustrated by the Dirac belt trick and fundamental to the antisymmetric nature of fermionic wavefunctions. In engineering, turns provide an intuitive measure for gear ratios in robotics and CNC machining, where fractional or multiple turns express the input-output relationship more accessibly than degrees; for instance, a 1.5-turn input equates to 540° and is common in multi-stage transmissions to balance torque and speed. In robotic systems, high gear ratios like 60:1 mean the motor completes 60 turns for one output revolution, enabling precise control in legged robots for dynamic maneuvers such as turning on low-friction surfaces. This unit avoids cumbersome degree multiples, facilitating programming of joint actuators in collaborative robots and automated machining paths. In navigation and aviation, headings are typically normalized modulo 360° (one turn) to represent direction consistently.) GPS-based RNAV procedures use track angle error (TAE) displays to improve accuracy during instrument approaches, as shown in studies on analog TAE aiding pilot performance. This practice enhances accuracy in GPS-based RNAV procedures, where fly-by waypoints anticipate turns to minimize deviation from the desired track. In mathematics education, the turn unit underpins advocacy for tau (τ=2π≈6.28\tau = 2\pi \approx 6.28) as the intuitive circle constant, representing one full rotation and simplifying formulas like arc length s=τrθs = \tau r \theta (with θ\theta in turns) over radian-based equivalents. Since the 2010s, this perspective has gained traction in pedagogical discussions, with tau promoted in resources to foster conceptual clarity for students learning trigonometry and geometry, as tau aligns directly with the periodicity of circular functions without the factor of 2. Textbooks and curricula increasingly reference tau for its alignment with full rotations, reducing misconceptions in angle measurements compared to pi-based conventions.

References

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