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gon
Compass graded with 400 gon
General information
Unit ofAngle
Symbolgon, ᵍ, grad
Conversions
1 gon in ...... is equal to ...
   turns   ⁠1/400⁠ turn
   radians   ⁠π/200⁠ rad
≈ 0.0157... rad
   milliradians   5π mrad
≈ 15.71... mrad
   degrees   0.9°
   minutes of arc   54′

In trigonometry, the gradian – also known as the gon (from Ancient Greek γωνία (gōnía) 'angle'), grad, or grade[1] – is a unit of measurement of an angle, defined as one-hundredth of the right angle; in other words, 100 gradians is equal to 90 degrees.[2][3][4] It is equivalent to ⁠1/400⁠ of a turn,[5] ⁠9/10⁠ of a degree, or ⁠π/200⁠ of a radian. Measuring angles in gradians (gons) is said to employ the centesimal system of angular measurement, initiated as part of metrication and decimalisation efforts.[6][7][8][a]

In continental Europe, the French word centigrade, also known as centesimal minute of arc, was in use for one hundredth of a grade; similarly, the centesimal second of arc was defined as one hundredth of a centesimal arc-minute, analogous to decimal time and the sexagesimal minutes and seconds of arc.[12] The chance of confusion was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.[13][14]

Gradians (gons) are principally used in surveying (especially in Europe),[15][7][16] and to a lesser extent in mining[17] and geology.[18][19]

The gon (gradian) is a legally recognised unit of measurement in the European Union[20]: 9  and in Switzerland.[21] However, this unit is not part of the International System of Units (SI).[22][20]: 9–10 

History and name

[edit]

The unit originated in France in connection with the French Revolution as the grade, along with the metric system, hence it is occasionally referred to as a metric degree. Due to confusion with the existing term grad(e) in some northern European countries (meaning a standard degree, ⁠1/360⁠ of a turn), the name gon was later adopted, first in those regions, and later as the international standard.[which?] In France, it was also called grade nouveau. In German, the unit was formerly also called Neugrad (new degree) (whereas the standard degree was referred to as Altgrad (old degree)), likewise nygrad in Danish, Swedish and Norwegian (also gradian), and nýgráða in Icelandic.

Although attempts at a general introduction were made, the unit was only adopted in some countries, and for specialised areas such as surveying,[15][7][16] mining[17] and geology.[18][19] Today, the degree, ⁠1/360⁠ of a turn, or the mathematically more convenient radian, ⁠1/2π⁠ of a turn (used in the SI system of units) is generally used instead.

In the 1970s –1990s, most scientific calculators offered the gon (gradian), as well as radians and degrees, for their trigonometric functions.[23] In the 2010s, some scientific calculators lack support for gradians.[24]

Symbol

[edit]
◌ᵍ
Gon
In UnicodeU+1D4D ᵍ MODIFIER LETTER SMALL G
Related
See alsoU+00B0 ° DEGREE SIGN

The international standard symbol for this unit is "gon" (see ISO 31-1, Annex B).[needs update] Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50g = 45°. A metric prefix is sometimes used, as in "dgon", "cgon", "mgon", denoting respectively 0.1 gon, 0.01 gon, 0.001 gon. Centesimal arc-minutes and centesimal arc-seconds were also denoted with superscripts c and cc, respectively.

SI multiples of gon (gon)
Submultiples Multiples
Value SI symbol Name Value SI symbol Name
10−1 gon dgon decigon 101 gon dagon decagon
10−2 gon cgon centigon 102 gon hgon hectogon
10−3 gon mgon milligon 103 gon kgon kilogon
10−6 gon μgon microgon 106 gon Mgon megagon
10−9 gon ngon nanogon 109 gon Ggon gigagon
10−12 gon pgon picogon 1012 gon Tgon teragon
10−15 gon fgon femtogon 1015 gon Pgon petagon
10−18 gon agon attogon 1018 gon Egon exagon
10−21 gon zgon zeptogon 1021 gon Zgon zettagon
10−24 gon ygon yoctogon 1024 gon Ygon yottagon
10−27 gon rgon rontogon 1027 gon Rgon ronnagon
10−30 gon qgon quectogon 1030 gon Qgon quettagon

Advantages and disadvantages

[edit]

Each quadrant is assigned a range of 100 gon, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles.

0° = 0 gradians
90° = 100 gradians
180° = 200 gradians
270° = 300 gradians
360° = 400 gradians

One advantage of this unit is that right angles to a given angle are easily determined. If one is sighting down a compass course of 117 gon, the direction to one's left is 17 gon, to one's right 217 gon, and behind one 317 gon. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (as ⁠33+1/3⁠ gon and ⁠66+2/3⁠ gon respectively).

Conversion

[edit]
Conversion of common angles
Turns Radians Degrees Gradians
0 turn 0 rad 0° 0g
⁠1/72⁠ turn ⁠π/36⁠ or ⁠𝜏/72⁠ rad 5° ⁠5+5/9⁠g
⁠1/24⁠ turn ⁠π/12⁠ or ⁠𝜏/24⁠ rad 15° ⁠16+2/3⁠g
⁠1/16⁠ turn ⁠π/8⁠ or ⁠𝜏/16⁠ rad 22.5° 25g
⁠1/12⁠ turn ⁠π/6⁠ or ⁠𝜏/12⁠ rad 30° ⁠33+1/3⁠g
⁠1/10⁠ turn ⁠π/5⁠ or ⁠𝜏/10⁠ rad 36° 40g
⁠1/8⁠ turn ⁠π/4⁠ or ⁠𝜏/8⁠ rad 45° 50g
⁠1/2π or 𝜏⁠ turn 1 rad approx. 57.3° approx. 63.7g
⁠1/6⁠ turn ⁠π/3⁠ or ⁠𝜏/6⁠ rad 60° ⁠66+2/3⁠g
⁠1/5⁠ turn ⁠2π or 𝜏/5⁠ rad 72° 80g
⁠1/4⁠ turn ⁠π/2⁠ or ⁠𝜏/4⁠ rad 90° 100g
⁠1/3⁠ turn ⁠2π or 𝜏/3⁠ rad 120° ⁠133+1/3⁠g
⁠2/5⁠ turn ⁠4π or 2𝜏 or α/5⁠ rad 144° 160g
⁠1/2⁠ turn π or ⁠𝜏/2⁠ rad 180° 200g
⁠3/4⁠ turn ⁠3π or ρ/2⁠ or ⁠3𝜏/4⁠ rad 270° 300g
1 turn 𝜏 or 2π rad 360° 400g

Relation to the metre

[edit]
Further information: History of the metre § Meridional definition
See also: Arc length § Great circles on Earth, and Nautical mile § Similar definitions
An early definition of the metre was one ten-millionth of the distance from the North Pole to the equator, measured along a meridian through Paris.

In the 18th century, the metre was defined as the 10-millionth part of a quarter meridian. Thus, 1 gon corresponds to an arc length along the Earth's surface of approximately 100 kilometres; 1 centigon to 1 kilometre; 10 microgons to 1 metre.[25] (The metre has been redefined with increasing precision since then.)

Relation to the SI system of units

[edit]

The gradian is not part of the International System of Units (SI). The EU directive on the units of measurement[20]: 9–10  notes that the gradian "does not appear in the lists drawn up by the CGPM, CIPM or BIPM." The most recent, 9th edition of the SI Brochure does not mention the gradian at all.[22] The previous edition mentioned it only in the following footnote:[26]

The gon (or grad, where grad is an alternative name for the gon) is an alternative unit of plane angle to the degree, defined as (π/200) rad. Thus there are 100 gon in a right angle. The potential value of the gon in navigation is that because the distance from the pole to the equator of the Earth is approximately 10000 km, 1 km on the surface of the Earth subtends an angle of one centigon at the centre of the Earth. However the gon is rarely used.

See also

[edit]

Notes

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Gradian

View on Grokipedia
from Grokipedia
The gradian, also known as the gon or grade, is a unit of angular measurement defined as one-hundredth of a , equivalent to 1/400 of a full circle. It measures angles in a centesimal system, where a equals 100 gradians and a complete equals 400 gradians. One gradian corresponds to 0.9 degrees or approximately 0.015708 radians (π/200). This unit facilitates decimal-based calculations, aligning with the 's emphasis on powers of ten, and subdivisions include centigradians (0.01 gradian), often used for precise measurements. The gradian originated in during the development of the in the late 18th and early 19th centuries, proposed as a rational alternative to the degree to promote decimal consistency in scientific and technical fields. It gained traction alongside other metric reforms, such as the centigrade temperature scale, but its adoption remained limited outside specific applications. Primarily used in , , and —especially in —the gradian simplifies right-angle divisions and decimal arithmetic for land measurement and mapping tasks. It appears in some geological and contexts, as well as on scientific calculators and software supporting angular computations, though degrees and radians dominate globally.

Definition and Fundamentals

Definition

The gradian, also known as the gon or grade, is a unit of plane defined as one four-hundredth of a full circle. This makes it a centesimal measure, where the entire is partitioned into 400 equal gradian units for angular quantification. In this system, a —or quadrant—corresponds precisely to 100 gradians, emphasizing its alignment with subdivisions. The gradian's structure thus divides the circle into parts that are multiples of 0.01 of a quadrant, promoting ease in decimal arithmetic for geometric computations.

Symbol and Notation

The gradian, serving as a decimal-based unit for plane angle measurement, employs specific symbols and notations in technical literature and standards. The international standard designates "gon" as the official name and symbol for the unit. In contemporary usage, particularly in mathematical and engineering contexts, the primary notation for expressing angles in gradians is a superscript "g" placed after the numerical value, analogous to the degree symbol; for example, a right angle is written as 100g100^\text{g}. This superscript form distinguishes gradian measurements from degrees while maintaining compact readability in formulas and diagrams. The unit symbol "gon" is used for the unit itself, while the superscript "g" denotes angles measured in gradians. Alternative notations include the abbreviations "gr" and "gon", which appear in various international texts and software implementations for compatibility and clarity. Historical variations trace back to early French developments, where the unit was termed "grade" and abbreviated as "grd" in older texts, reflecting its origins in proposals. The (ISO) established "gon" as the preferred symbol in ISO 80000-3:2019 to promote uniformity across languages and avoid ambiguity with other terms like "grad" for .

Historical Development

Origins and Etymology

The gradian, also known as the grade or gon, emerged from efforts by the in the 18th and 19th centuries to reform angular measurement as part of the broader overhaul, aiming to replace the cumbersome divisions of the circle (based on 360 degrees) with a purely for simplified calculations in science and engineering. Early proposals during the sought a universal, rational framework tied to natural phenomena, much like the metre's basis in Earth's meridian; the for angles was introduced by the law of 11 Brumaire Year IV on 1 November 1795, where the right angle equaled 100 grades, dividing the full circle into 400 grades to align with base-10 arithmetic. This work emphasized the practical benefits of decimal subdivisions for fields like astronomy and , where traditional degrees complicated computations. In 1897, a commission including advocated for the system's adoption, highlighting its advantages for calculations without needing two-digit multiplications in conversions. Etymologically, the term "grade" derives from the French "grade," meaning a step or degree, reflecting the unit's conception as incremental divisions akin to steps in a decimal progression. To promote linguistic neutrality and avoid confusion with the English "grade" denoting or incline, the name evolved to "gon" in the , drawn from the Greek "gōnia" (γωνία), signifying corner or angle, paralleling its use in terms like "."

Adoption and Decline

The gradian experienced limited adoption primarily in European surveying contexts. It was employed in French surveying practices until the mid-20th century, aligning with the country's metric reforms and facilitating decimal-based angular calculations in and mapping, as well as in Swiss systems, where the gon appears in official projection formulas for coordinate transformations. The unit's inclusion in ISO standards, such as ISO 80000-1:2009 for general quantities and units and ISO/IEC 13249-3:2016 for information technology data types, recognizes it as a valid plane angle measure but renders it non-mandatory alongside the preferred . The gradian's decline stemmed from the dominant tradition of the degree unit in , , and international , where compatibility with historical tables and instruments favored the system. Post-1970s computational developments further entrenched this shift, as early digital software and calculators were predominantly programmed for degrees, creating inertia against adopting the gradian despite its decimal advantages. By the late , it had become largely obsolete outside niche European applications, supplanted by degrees for broader .

Conversions and Mathematical Relations

Formulas for Conversion

The gradian, also known as the gon, is defined such that a full circle corresponds to 400 gradians, providing a basis for conversions to other angular units. This equivalence stems from the unit's design, where 400 gradians equal 360 degrees and 2π radians. To convert gradians to degrees, the formula is derived by dividing the full-circle values: degrees = gradians × (360/400) = gradians × 0.9. Thus, 1 gradian = 0.9 degrees. The inverse conversion is gradians = degrees × (400/360) = degrees × (10/9). For conversion to radians, the relation follows from the full-circle equivalences: radians = gradians × (2π/400) = gradians × (π/200). Therefore, 1 gradian = π/200 radians, approximately 0.01570796 radians. The bidirectional formula is gradians = radians × (400/(2π)) = radians × (200/π). These formulas reflect the gradian's alignment with a structure for angular measurement, facilitating calculations in systems preferring base-10 divisions.

Equivalences with Other Units

The gradian, denoted as gon, equates to one-fourth of a , making a full circle 400 gradians, which corresponds exactly to 360 degrees, 2π radians, and 1 turn. Similarly, a measures 100 gradians, equivalent to 90 degrees, π/2 radians, and 0.25 turns. These relations stem from the gradian's centesimal basis, dividing the circle into 400 equal parts for alignment with systems. In comparisons to subdivisions, 1 gradian equals 0.9 degrees and thus 54 arcminutes, while 1 degree approximates 1.111 gradians (precisely 10/9 gradians). One gradian further subdivides to 3240 arcseconds. Such equivalences facilitate in fields like , where gradians align with metric precision. The following table summarizes breakdowns of a full circle across key units, including percentages for proportional representation:
DescriptionGradians (gons)Degrees (°)Radians (rad)Percentage of Circle (%)
Full Circle4003602π100
(Quadrant)10090π/225
1 Degree1.111... (10/9)1π/1800.277... (1/360)
1 Turn4003602π100
Conversion methods between gradians and other units are detailed in prior sections.

Advantages, Disadvantages, and Practical Use

Benefits and Drawbacks

The gradian aligns closely with the decimal nature of the , enabling simpler arithmetic operations when performing calculations involving angles, as subdivisions avoid the sexagesimal fractions common in degree-based systems. This decimal compatibility particularly benefits computations in and scientific contexts where base-10 metrics predominate. A key theoretical strength lies in its precise division of right angles into 100 gradians exactly, facilitating straightforward partitioning for perpendicular measurements without residual fractions. For instance, angles like 45 degrees convert directly to 50 gradians, minimizing compared to the 45/360 simplification required in degrees. Despite these merits, the gradian lacks the intuitive appeal of the degree system, rooted in millennia of cultural and historical use of the 360-degree circle, which hinders quick mental estimation and visualization for most users. Moreover, standard and tables, optimized for degrees or radians, yield non-integer values for many geometrically significant angles in gradians—such as 30 degrees equating to 331333\frac{1}{3} gradians and 60 degrees to 662366\frac{2}{3} gradians—complicating derivations and applications outside specialized frameworks.

Applications in Surveying and Engineering

In , the gradian is employed in metric cadastral surveys, particularly in and , where it enables decimal representations of bearings and directions for land boundary delineation and property mapping. This unit aligns with the decimal nature of the , simplifying computations in geodetic work such as traverse closures and alignment projections. For instance, in French land surveying practices, theodolites calibrated in gradians were standard in the for measuring horizontal and vertical angles with centesimal precision, supporting national cadastral maintenance under the Direction Générale des Impôts. Similarly, Switzerland's federal cadastral system recognizes the gon (gradian) as a legal unit, facilitating its use in official topographic and boundary surveys conducted by the Federal Office of Topography. In , gradians are utilized for angle inputs in design and construction tasks, including road alignments, bridge layouts, and tunnel , where decimal divisions aid in iterative calculations for right- integrations. Software tools compliant with ISO standards, such as , incorporate a gon mode for angle measurements, allowing engineers to specify directions in gradians for compatibility with European data as of 2025. This support extends to ISO 80000-recognized applications, ensuring in geodetic and workflows. A niche application persists in ballistic engineering within certain military contexts, such as French artillery systems, where gradians facilitate corrections and firing table computations due to their alignment with metric decimal scaling. Overall, these uses highlight the gradian's role in technical fields emphasizing decimal precision over traditional sexagesimal systems.

Relation to Metric and SI Systems

Connection to the Metre

The gradian, as a decimal-based unit of angular measurement, integrates seamlessly with the in metric by enabling straightforward decimal ratios in . For instance, in a right-angled with legs each measuring 1 , the acute angles measure exactly 50 gradians, where tan⁡(50g)=1\tan(50^\text{g}) = 1, representing a precise decimal of opposite to adjacent sides without fractional complications inherent in degree-based systems. This alignment supports conceptual clarity in geometric computations involving decimal lengths, as the centesimal division (100 gradians per ) mirrors the decimal structure of the , facilitating calculations in fields requiring precise spatial relationships. In and , the gradian enhances metric triangulation by allowing s to be computed directly in metres using angular values, bypassing the conversions required in (degree-minute-second) systems. For example, when determining the along a curved boundary or geodetic line, the s=r⋅θg⋅π200s = r \cdot \theta^\text{g} \cdot \frac{\pi}{200} (where rr is the in metres and θg\theta^\text{g} is the angle in gradians) permits inputs for both angular and linear components, simplifying fieldwork and reducing errors in large-scale mappings. This practical integration was particularly valued in early 20th-century French military and topographic surveys, where gradians expressed latitudes and longitudes alongside metric coordinates for efficient border and terrain delineations. Historically, the gradian emerged as the proposed "metric angle" during the late 18th-century French metrication efforts, designed to complement the by establishing a uniform framework for angular and linear measurements in . Originating from the ' 1791 reforms, it aimed to decimalize angular units alongside lengths, envisioning a cohesive system where geodetic computations—such as determining Earth's or plotting projections—could proceed entirely in base-10 without mixing with the Babylonian-derived divisions. Although not universally adopted, this tie underscores the gradian's role in pursuing harmony between angles and the for scientific precision.

Status within the SI Framework

The gradian, also known as the gon, holds a non-SI status within the (SI), where it is recognized as a derived unit of plane angle rather than a base unit, and is not listed among the accepted non-SI units for use with the SI as outlined in the 9th edition of the SI Brochure (2019). This edition, reflecting the 2019 revision of the SI, removed the gradian from prior mentions in earlier brochures, affirming its non-preferred position while maintaining its validity as a supplementary unit expressible in coherent SI terms. Its decimal alignment supports compatibility with the metric system's emphasis on base-10 divisions, though this does not confer formal SI endorsement. The gradian is fully compatible with SI principles, as it can be precisely expressed in radians—the SI coherent derived unit for plane angle—with the relation 11 gon =π200= \frac{\pi}{200} rad, ensuring no dimensional conflicts or inconsistencies in calculations. Despite this , the is explicitly preferred for scientific and technical applications to promote coherence within the SI framework. Regulatory guidance from the (ISO) further clarifies the gradian's role, with ISO 80000-3:2006 defining it as a unit of plane angle equivalent to 1200\frac{1}{200} rad and permitting its use in technical contexts, particularly where decimal angular measures facilitate computations aligned with metric lengths. The 2019 SI revision reinforces this non-preferred but permissible status, allowing the gradian in specialized fields without undermining the radian's primacy.

References

  1. https://mathworld.wolfram.com/Gradian.html
  2. https://support.esri.com/en-us/gis-dictionary/gradian
  3. https://www.ck12.org/flexi/algebra-ii/conversion-between-degrees-and-radians/what-is-the-centesimal-system-of-measuring-angles/
  4. https://www.yourdictionary.com/gradian
  5. https://metricsystem.net/non-si-units/other-non-si-units/gradian/
  6. https://www.automation.com/en-us/articles/2013-1/do-you-know-your-arc-seconds-from-your-gradians
  7. https://artofproblemsolving.com/wiki/index.php/Gradian
  8. https://www.examples.com/physics/gradian.html
  9. https://www.metric-conversions.org/angle/gradians-conversion.htm
  10. https://www.xconvert.com/unit-converter/degrees-to-gradians
  11. https://graphsearch.epfl.ch/concept/201629
  12. https://www.iso.org/standard/64974.html
  13. https://www.sizes.com/units/gon.htm
  14. https://www.ibiblio.org/units/dictG.html
  15. https://mathshistory.st-andrews.ac.uk/HistTopics/Decimal_time/
  16. https://www.etymonline.com/word/-gon
  17. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8
  18. https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication1038.pdf
  19. https://mathresearch.utsa.edu/wiki/index.php?title=Lines_%26_Angles
  20. https://home.adelphi.edu/~bradley/Modules/Trigonometry.pdf
  21. https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf
  22. https://www.learner.org/series/learning-math-measurement/classroom-case-studies-3-5/part-b-reasoning-about-measurement-40-minutes/
  23. https://prestashop.hydrografie.synology.me/22-angle
  24. https://help.autodesk.com/view/ACD/2025/ENU/?guid=GUID-FD3A36E1-FC7F-43CB-B7B5-1806FAAC12B2
  25. https://www.calcsimpler.com/units-and-measures/gradian-gon-metric-angle-unit
  26. https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_68.pdf
  27. https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507
  28. https://physics.nist.gov/cuu/pdf/sp811.pdf
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