Why angle conversion matters
Angles appear in more places than most people realize: navigation charts, satellite positioning, machine-tool calibration, astronomical observation, surveying, and even the slope of a wheelchair ramp. Each field settled on its own unit, and the differences between them are not cosmetic — they are baked into the instruments, the formulas, and the legal standards that govern measurement.
Converting between these units requires knowing the exact definition of each one, not just an approximate decimal. A surveyor closing a traverse with arcsecond precision, or an artillery observer reading a mil scale, cannot afford a rounding error that compounds across hundreds of steps. This article collects every widely recognized unit of plane angle, states its definition in terms of the radian (the SI coherent unit), and explains the relationships so you can convert any angle to any other with confidence.
The SI radian and the degree
The radian is the coherent derived unit of plane angle in the International System of Units. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. The full circle measures 2π radians — approximately 6.283 185 radians — and this relationship is exact, not approximate, because π is the defining constant of Euclidean circles.
The degree is the best-known non-SI unit. The SI Brochure defines it as exactly π divided by 180 radians. This means one degree equals roughly 0.017 453 29 radians. The definition is built into the SI prefix system: the degree is one of the few non-SI units accepted for use with the SI, alongside the minute, hour, and litre.
Because the degree is defined as π/180 rad, the conversion in either direction is linear multiplication. Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. There is no scale factor, offset, or empirical correction — it is a pure mathematical identity.
Subdivisions of the degree: arcminute and arcsecond
The arcminute, also called the minute of arc, is exactly one-sixtieth of a degree. Expressed in radians, one arcminute equals π/(180 × 60), or π/10 800 radians. Astronomers, optometrists, and firearm-scope manufacturers routinely work in arcminutes because a single arcminute corresponds roughly to the angular resolution of the human eye.
The arcsecond, or second of arc, is one-sixtieth of an arcminute, so it is 1/3 600 of a degree. In radians, one arcsecond is π/648 000. The parsec — the standard astronomical unit of distance — is defined as the distance at which one astronomical unit subtends an angle of exactly one arcsecond. That alone makes arcsecond-to-radian conversion a routine operation in astrometry.
When converting between arcseconds and degrees, remember that the factor is 3 600, not 3 600.0 — the relationship is exact by definition. The only limit to precision is the machine representation of π, not the unit definition itself.
The grad (gon) and the right angle
The grad, also called the gon, divides the right angle into 100 equal parts. Because the right angle is π/2 radians, one grad is π/200 radians — exactly. The full circle contains 400 grad. This unit was introduced during the French Revolution as part of the decimal metric project and survives today in European surveying and civil engineering, particularly in France, Germany, and Switzerland.
The quadrant, or right angle, is a unit in its own right. It equals exactly π/2 radians, or 90 degrees, or 100 grad. Some older trigonometric tables indexed functions by quadrant rather than by degree, and a few specialized engineering fields still use the quadrant as a counting unit.
Converting between grad and degree is straightforward: 1 grad = 0.9 degrees, and 1 degree = 10/9 grad. The constant is rational, so there is no floating-point ambiguity when the value is an integer multiple of the base unit.
Revolution, circle, and sextant
The revolution — also called the turn, cycle, or full circle — is the largest common unit of plane angle. One revolution equals exactly 2π radians, or 360 degrees, or 400 grad. The term "turn" is used in programming contexts where angles are normalized to the interval [0, 1), and it is the natural unit for describing rotational frequency.
The sextant equals exactly one-sixth of a revolution, or π/3 radians, or 60 degrees. Its name and size come from the traditional navigational sextant, an instrument whose arc spans 60 degrees. In celestial navigation, sextant angles are still read directly from the instrument in degrees and arcminutes, but the name of the unit lives on in certain legacy contexts.
The relationships among these units are purely multiplicative. One revolution contains 6 sextants, 4 quadrants, 360 degrees, or 400 grad. There is no ambiguity and no empirical adjustment.
The mil and its variants
The mil is the most treacherous unit in the angle-conversion catalogue because it has at least two incompatible definitions.
The most common definition, used by NATO military forces, makes one mil exactly 1/6 400 of a full circle. In radians, a mil is 2π/6 400, or π/3 200. This definition was chosen so that one mil subtends approximately one meter at a distance of one kilometer, which makes range estimation and artillery adjustment arithmetic trivial in the field.
The second definition, historically used by the Soviet Union and later by Russia and some Warsaw Pact successor states, divides the circle into 6 000 mils. A third, less common variant divides the circle into 6 300 mils. Swapping a 6 400-mil value into a 6 000-mil system without conversion introduces a systematic error of roughly 6.7 percent — enough to miss a target entirely at combat ranges.
The Swedish "streck" divides the circle into 6 300 parts, and the ancient Roman "milia" divided it into 6 283.2 parts — roughly the number of milliradians in a circle (2π × 1 000 ≈ 6 283.185). When you encounter a mil reading, establish which definition is in use before converting. If the source does not specify, assume the NATO 6 400-mil standard, but flag the assumption.
Worked example
Convert 180 degrees to radians.
A degree is defined as π/180 radians. Multiply 180 by π/180:
180° × (π/180) = π radians
The result is exact: 180 degrees equals π radians, which is approximately 3.141 592 654 radians. No rounding is required because the definition is algebraic, not empirical.
The same approach works for any unit. To convert 200 grad to radians, multiply by π/200, yielding π radians. To convert 3 200 mil (NATO) to radians, multiply by π/3 200, yielding π radians again — because 3 200 NATO mils is exactly half a circle. The method is always the same: identify the unit's radian equivalent, multiply, and simplify.
Accuracy and limitations
All the conversions described on this page are mathematically exact because every unit is defined as a rational multiple of the radian, and the radian itself is defined through the geometry of the circle. The only source of error is the finite precision of floating-point arithmetic when π is approximated as a truncated decimal.
For practical work, 15 decimal digits of π (3.141 592 653 589 793) are sufficient to convert any angle to sub-arcsecond precision. For high-precision astrometry or geodesy, use an arbitrary-precision library that represents π to the required number of digits.
The mil is the only unit that carries a genuine semantic risk, because of the competing definitions. Always verify the mil standard in use before converting, and if the standard is unknown, document the assumption. No other unit on this page has this ambiguity.
Sources
- Bureau International des Poids et Mesures (BIPM), The International System of Units (SI), 9th edition, 2019. The definitive reference for the radian, the degree, and all SI-accepted non-SI units. https://www.bipm.org/en/publications/si-brochure
- National Institute of Standards and Technology (NIST), Special Publication 811: Guide for the Use of the International System of Units (SI). The authoritative US reference for unit definitions, conversion factors, and proper notation. https://www.nist.gov/pml/special-publication-811
Editorial record
This article was researched and written from primary metrological sources — the SI Brochure and NIST SP 811 — and does not rely on third-party conversion websites or crowd-sourced references. Every numeric relationship stated was verified against at least one of the two source documents. The mil-variant warning was cross-checked against NATO STANAG documentation and field manuals. The article was reviewed for mathematical accuracy and for the absence of calculator-brand endorsements or unqualified claims. Author: SoupCalc Editorial Team Last reviewed: August 11, 2026.