Angle conversion family

Angle unit conversions

Plane angle is a derived quantity treated within the SI as a quantity with the unit one; the radian (rad) is its coherent SI unit. This family uses the degree as its conversion base for compatibility with the existing calculators.

Five named units

Unit and factor reference

Base: °

Factors are stored as decimal strings relative to degree (°).
Unit	Symbol	Factor to base	Relationship	Definition and variant
degree	`°`	`1`	conventional	The degree is a long-standing non-SI plane-angle unit with the exact relation 1° = π/180 rad; a full turn is 360°. Named variant: Degree of plane or phase angle; not degree Celsius or degree Fahrenheit. The base factor is exactly 1 by the family base-unit choice. Conversion to radians involves π and therefore has no finite decimal expansion.
radian	`rad`	`57.2957795131`	approximate	The radian is the coherent SI unit for plane angle. One radian is the angle subtended by an arc equal in length to the radius; equivalently, a right angle is π/2 rad and 1 rad = 180/π degrees. Named variant: Coherent SI unit of plane angle; rad can also name the obsolete absorbed-dose unit in unrelated contexts. The exact degree factor is 180/π ≈ 57.29577951308232. The preserved baseFactor 57.2957795131 is rounded and is about 1.77×10⁻¹¹ degree higher per radian.
gradian	`gon`	`0.9`	defined	A gradian (gon) divides a full turn into 400 parts, so 1 gon = 0.9° exactly. Named variant: Plane-angle gradian/gon; not a percentage grade or academic grade. The terminating degree relationship is exact.
turn	`turn`	`360`	conventional	A turn is one complete plane rotation, equal to 360° or 2π rad. Named variant: Complete plane rotation; the display token 'turn' is used by CalcMotive rather than asserting a standardized international symbol. The relationship 1 turn = 360° is exact.
arcminute	`arcmin`	`0.016666666666666666`	approximate	An arcminute is one minute of plane angle: exactly 1/60 degree or π/10,800 rad. Named variant: Minute of plane angle; not minute of elapsed time. The exact degree factor 1/60 is repeating in decimal. The source dataset preserves JavaScript's raw evaluated value; the generator's 15-significant-digit emission is 0.0166666666666667 degree.

Every ordered pair

All 20 angle calculators

From degrees (°)

From radians (rad)

Radians to Degreesrad → °
Radians to Gradiansrad → gon
Radians to Turnsrad → turn
Radians to Arcminutesrad → arcmin

From gradians (gon)

From turns (turn)

Turns to Degreesturn → °
Turns to Radiansturn → rad
Turns to Gradiansturn → gon
Turns to Arcminutesturn → arcmin

From arcminutes (arcmin)

Arcminutes to Degreesarcmin → °
Arcminutes to Radiansarcmin → rad
Arcminutes to Gradiansarcmin → gon
Arcminutes to Turnsarcmin → turn

Evidence and review status

This family is machine-reviewed-pending-human. Automated checks verify structure, factor parity and source references; they do not constitute human or specialist approval.

Family assumptions and limits

Signed values are supported, but positive/negative rotational direction depends on the user's coordinate convention.
This family performs linear unit conversion only; it does not normalize angles to [0, 360), (−180, 180], or another interval.
Radian and degree are explicitly distinct: one right angle is π/2 rad and 90°, so 1 rad = 180/π degrees and 1° = π/180 rad.

Authoritative references

The International System of Units (SI), 9th edition, version 4.01; Bureau International des Poids et Mesures; Sections 2.3.3 to 2.3.4, pp. 132 to 134: plane angle, radian, and right angle = π/2 rad; Section 4, Table 8, p. 140: 1° = π/180 rad and 1′ = 1/60 degree.; accessed 2026-08-07
NIST Guide to the SI, Appendix B.8: Factors for Units Listed Alphabetically; National Institute of Standards and Technology; Entries 'degree (angle)', 'gon (also called grade)' = 0.9 degree, and 'revolution' = 2π rad; factors support the degree, gradian, and full-turn relationships.; accessed 2026-08-07

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