Angle Converter
Convert degrees, radians, gradians, arcminutes and arcseconds — and keep the four incompatible military ‘mils’ apart, because NATO, Soviet, Swedish and true milliradian all differ.
Angle Converter
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What this result does not account for
- The word mil is ambiguous — four definitions are in live use and this engine cannot know which your equipment follows.
- Mil-ranging assumes the small-angle approximation and an accurately known target dimension; both break down at large angles.
- Angular mil, minute of angle and the thou of length are three different things that share the name mil.
- Bearings depend on the north reference used — true, grid or magnetic — which is a separate correction from unit conversion.
In short: Angle conversion routes through the radian, the SI coherent unit, where a full turn is exactly 2π. The trap is the mil: a true milliradian gives 6,283.185 per circle, but NATO uses 6,400, the former Soviet bloc 6,000 and Sweden 6,300. The NATO mil is 1.825% smaller than a true milliradian, and all four are still in live equipment.
Formula
Each factor is the number of radians in one of that unit. The radian is the ratio of arc length to radius and is therefore dimensionless, which is why it is the natural unit for calculus and why every trigonometric function in software expects radians rather than degrees.
Worked Example
- Identify the source factor. One degree is π/180 = 0.0174532925 radians.
- Convert to the SI unit. 1 × 0.0174532925 = 0.0174532925 rad.
- Identify the target factor. One radian is 1 rad.
- Divide. 0.0174532925 ÷ 1 = 0.017453293 rad.
- Check the mil. The same angle is 17.777778 NATO mils but 17.453293 true milliradians — not the same number.
Why the mil exists. At small angles the tangent of an angle is very nearly the angle itself in radians, so one milliradian subtends almost exactly one metre at one thousand metres. That turns an angular measurement in a sight into a range estimate with nothing but division: range in metres equals target size in millimetres divided by the mils it covers. NATO rounded 6,283.185 to 6,400 because it divides evenly by eight, which mattered when fire-control arithmetic was done by hand.
Strengths & Limits Of This Model
Where this engine is strong
- Keeps all four military mils separate instead of pretending there is one.
- Shows the exact subtension and the small-angle approximation side by side.
- Exact sexagesimal DMS breakdown with no rounding into 60.
- Every factor derived from the definition of the radian.
Where it stops
- Cannot resolve which mil convention a given sight or fire-control system uses.
- Does not handle bearing arithmetic or north references.
Practical Use Cases
Surveying and navigation
Bearings are recorded in degrees, minutes and seconds in most legal and cadastral records, but computed in radians and sometimes plotted in gradians on continental European equipment. The DMS readout keeps the sexagesimal breakdown exact.
Optics and marksmanship
Scope turrets are graduated in true milliradians or in minutes of angle. A scope marked MRAD and a fire-control system marked in NATO mils are 1.825 per cent apart, which is a real miss at distance.
Programming and engineering mathematics
Every trigonometric function in a standard library takes radians. Converting degrees at the boundary of a calculation is routine — see the Frequency Converter for the closely related 2π factor in angular frequency.
Astronomy
Angular separations are quoted in arcminutes and arcseconds: the Moon spans about 31 arcminutes, and atmospheric seeing is measured in fractions of an arcsecond.
Methodology & Editorial Standards
All conversions route through the radian, defined as the angle subtending an arc equal in length to the radius. Because it is a ratio of two lengths the radian is dimensionless, and a full turn is exactly 2π. The degree, arcminute and arcsecond follow the Babylonian sexagesimal division and are exact fractions of the turn; the gradian divides the right angle into 100 parts. The four angular mils are given separately and explicitly rather than collapsed into one row, because they are genuinely different units: the true milliradian at 2,000π per circle, the NATO mil at 6,400, the Soviet mil at 6,000 and the Swedish streck at 6,300. The subtension readout computes the exact chord using the tangent and reports the small-angle linear approximation alongside it, so the point at which the approximation stops holding is visible rather than assumed. All conversion factors are exact by definition under the International System of Units, or exact by international agreement where the unit is defined by treaty. Values are held at full IEEE-754 double precision internally and rounded only for display, so chained conversions do not accumulate drift.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Angle Converter — 10 Expert FAQs
10 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
How many degrees are in a radian?
Exactly 180/π, which is 57.295780 degrees. The relationship is exact because it is definitional rather than measured: a full turn is 2π radians and 360 degrees, so the ratio follows directly. Going the other way, one degree is π/180 or 0.017453293 radians.
What is the difference between a mil and a milliradian?
A true milliradian is one thousandth of a radian, giving 6,283.185 per circle. A NATO mil is one 6,400th of a circle, chosen because it divides evenly and makes field arithmetic quick. The NATO mil is about 1.825 per cent smaller. The former Soviet bloc used 6,000 per circle and Sweden 6,300, so the word mil alone is ambiguous without knowing the equipment.
Why do mils subtend one metre at one thousand metres?
Because at small angles the tangent of an angle is very nearly equal to the angle measured in radians. One milliradian is 0.001 radians, so at a range of 1,000 metres it covers 1,000 × 0.001 = 1 metre. That lets a shooter or observer convert an angular measurement into a range with a single division and no trigonometry.
What is a gradian?
Also called a gon or a grade, it divides the right angle into 100 parts, so a full circle is 400 gradians. It was introduced alongside the metric system to decimalise angle measurement and survives mainly in European surveying and on scientific calculators, where it usually appears as a third mode beside degrees and radians.
How do I convert decimal degrees to degrees, minutes and seconds?
Take the whole number of degrees, multiply the remaining fraction by 60 to get arcminutes, then multiply that remainder by 60 again for arcseconds. So 1.5 degrees is 1° 30' 0". The system is base 60, inherited from Babylonian astronomy, which is also why the hour divides the same way.
Is a mil the same as a MOA?
No. A minute of angle is one sixtieth of a degree, so there are 21,600 in a circle. One true milliradian is 3.4377468 MOA. Both are used on rifle scopes and mixing them is a classic error — which is why the standard advice is never to pair a mil reticle with MOA turrets.
Why do trigonometric functions in code use radians?
Because the radian makes the calculus work cleanly: the derivative of sine is cosine only when the angle is in radians, and the series expansions that libraries actually compute assume it. Feeding degrees into a sine function without converting is one of the most common numerical bugs in graphics and simulation code.
What is an arcsecond used for?
Very small angular separations, chiefly in astronomy and precision surveying. One arcsecond is 1/3600 of a degree, about 4.85 microradians. The parsec is defined from it: the distance at which one astronomical unit subtends one arcsecond.
What is a point in navigation?
One thirty-second of a circle, or 11.25 degrees. The compass rose divides into 32 points — north, north by east, north-northeast and so on — which was the standard way of giving direction under sail before graduated compasses became universal.
Does the subtension figure use the exact formula or the approximation?
The main figure is exact, computed as twice the range times the tangent of half the angle. The linear small-angle approximation is shown beside it so you can see where the two agree. For angles of a few milliradians they match to six significant figures; for large angles the approximation breaks down visibly, which is why mil-ranging is only used on small angular targets.