Angle Converter: Degrees, Radians, Gradians and Military Mils

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How to use this tool

Enter an angle, choose the unit you have and the one you want. All ten units appear in the table at once, grouped into common, survey and military, because those three worlds use genuinely different divisions of the circle.

If you are working with a European total station, the unit you want is probably the gradian, also called the gon. A full circle is 400 gradians, so a right angle is exactly 100. That is the whole appeal: quadrants come out as round numbers.

What the conversion is actually doing

Everything is stored as degrees and converted by a multiply and a divide. Only the radian involves an irrational factor, since one radian is 180 divided by pi degrees. Everything else is a clean ratio, because every other unit is defined as some whole number of divisions of a full turn.

That is the useful way to think about the whole table. A turn is 360 degrees, 400 gradians, 6400 NATO mils, 6300 Swedish streck, 6000 Warsaw Pact mils, or 2 pi radians. Pick your divisor.

The mil is not a milliradian

This is the one that costs people money. A true milliradian divides the circle into 2000 pi, which is 6283.19 parts. Nobody wants to work with that, so artillery systems rounded it. NATO rounded up to 6400, the Warsaw Pact rounded down to 6000, and Sweden settled on 6300.

So a NATO mil is 0.9817 milliradians, not one. The 1.85 percent difference is invisible at short range and very visible at long range. Treating a scope adjustment in mils as if it were milliradians, or the reverse, puts you out by roughly one part in fifty, which at 1000 metres is about 1.8 metres.

The 6000 and 6400 systems are further apart still, about 6.7 percent. Equipment from different origins genuinely does not interoperate on this, and the tool keeps all three separate rather than pretending they are the same unit.

A worked check you can repeat on paper

A bearing recorded as 45 degrees. In gradians that is 50, since a gradian is nine tenths of a degree. In NATO mils it is 45 divided by 0.05625, which is 800. In radians it is pi over 4, or 0.785398.

Now the check that proves the system is consistent. A full turn should return whole numbers in every divisor based unit. Put 1 in with turn selected: 360 degrees, 400 gradians, 6400 NATO mils, 6300 streck, 6000 Warsaw Pact mils. No rounding anywhere.

Mistakes that catch people out

Treating mils as milliradians. Almost 2 percent of error. Fix: check whether your instrument uses 6400, 6300 or 6000 to the circle, and convert rather than assume.

Mixing gradians and degrees on a total station. A right angle reads 100 in one and 90 in the other, which looks obviously different, but 45 versus 50 does not. Fix: check the instrument setting before recording, not after.

Forgetting that arcminutes are sixtieths. Angle is not decimal in the common system. 30 arcminutes is half a degree, not three tenths. Fix: convert through the tool rather than moving the decimal point.

Using degrees in a trigonometric function that expects radians. Every programming language and spreadsheet defaults to radians. Fix: convert first, and remember the factor is 180 over pi, not 180 over 2 pi.

How this was checked

Every divisor was checked as a whole number rather than an approximation. One turn returns exactly 360 degrees, exactly 400 gradians, exactly 6400 NATO mils, exactly 6300 streck and exactly 6000 Warsaw Pact mils. One degree returns exactly 3600 arcseconds. One radian returns 57.29577951308232 degrees, which is 180 over pi to full double precision.

The milliradian to NATO mil ratio was checked at 1.0185916, confirming the two are not interchangeable. All 100 unit pairs were round tripped with drift at floating point level.

Related Calculators and Reference

For slopes expressed as percentages or ratios rather than angles, use the Grade to Degrees Converter. In structural geology the Apparent and True Dip Converter handles the projection problem, and the Stereonet plots planes and poles. For borehole work the Dogleg Severity Calculator works in degrees per length. Definitions follow the NIST Office of Weights and Measures. Also see the Trig Calculator: sin, cos and tan of the converted angle, with exact values.