Why this matters
Rotating a turret by 3.5 degrees is intuitive, but a navigation system expecting radians will silently produce a trajectory error of several kilometers over a long route. Different disciplines have settled on different angular units over centuries — mathematicians use radians because calculus formulas only work cleanly in that system, surveyors use gradians for their convenient base-100 subdivision, and military operators use NATO mils because one mil subtends roughly one meter at one kilometer. Switching between these systems manually involves multiplying by awkward constants like pi over 180, and a single arithmetic mistake cascades through every downstream calculation.
This converter handles all seven common angular units in a single pass: degrees, radians, gradians, arcminutes, arcseconds, revolutions, and NATO mils. You enter one value and one source unit, and every equivalent appears instantly with a copy button beside each result. The entire computation pivots on degrees as the intermediate currency, so every conversion path is exact and consistent rather than approximated through chained factor pairs.
All seven units at a glance
| Unit | Symbol | Full circle | Primary domain |
|---|---|---|---|
| Degrees | deg | 360 | Everyday geometry, engineering |
| Radians | rad | 2 pi (~6.283) | Mathematics, physics, calculus |
| Gradians | gon | 400 | Surveying, European engineering |
| Arcminutes | ' | 21,600 | Astronomy, optics |
| Arcseconds | " | 1,296,000 | Astronomy, precision instruments |
| Revolutions | rev | 1 | Rotational mechanics, motors |
| NATO mils | mil | 6,400 | Artillery, sniping, military targeting |
How to use it
Enter an angle value in the input field — positive or negative, integer or decimal.
Select the source unit from the dropdown (degrees, radians, gradians, or any of the other four).
View all seven equivalent values as they appear instantly below the input.
Click the copy icon next to any result to grab it for your spreadsheet or code editor.
Testing your result
Start with a full circle: enter 360 in degrees and confirm that radians show approximately 6.283, gradians show 400, revolutions show 1, and NATO mils show 6400. Then test a right angle: 90 degrees should yield pi over 2 radians (about 1.5708), 100 gradians, and 1600 NATO mils. For a sub-degree test, enter 1 arcminute and verify the degrees result is 0.016667. These spot checks confirm that the conversion factors through all intermediate paths are correct and that the tool handles both integer and fractional values precisely.
Common mistakes
Confusing NATO mils (6400 per circle) with the Swedish or Soviet mil (6000 or 6300 per circle), which produces a small but consequential targeting error at long range.
Treating arcminutes and arcseconds as subdivisions of radians rather than degrees — they are 1/60 and 1/3600 of a degree respectively, not a radian.
Forgetting that negative angles wrap around differently in modular arithmetic — entering -90 degrees is equivalent to 270 degrees in a positive direction.
Edge cases and limitations
The converter treats all values as scalar angles and does not perform modular wrapping — entering 400 degrees will display 400 degrees, not 40 degrees plus one revolution. For applications requiring normalized angles between 0 and 360 or 0 and 2 pi, you will need to apply the modulo yourself after copying the result. Very small values in arcseconds (below 0.001) may lose precision due to floating-point representation in the browser. All computation runs locally with no data leaving your device.
Real-world use cases
A robotics engineer converting joint angles from degrees in the CAD model to radians for the inverse kinematics library.
A surveyor translating a theodolite reading in gradians to decimal degrees for input into GIS software that only accepts the latter.
A physics student converting angular velocity from revolutions per minute to radians per second for a rotational dynamics homework problem.
Frequently asked questions
Q: Why are there 360 degrees?
A: Babylonian legacy — they used base-60 math and 360 was convenient (divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, and many more). It is also close to the number of days in a year, which made it practical for early astronomy.
Q: What is a radian?
A: The angle subtended by an arc equal in length to the radius. Exactly 2 pi radians equal 360 degrees. Radians are used in mathematics because trigonometric derivatives (like d/dx sin(x) = cos(x)) only hold in radians.
Q: What is a gradian?
A: One gradian is 1/400 of a full circle, so 100 gradians equal a right angle (90 degrees). Each gradian is 0.9 degrees. Gradians are used in surveying and some European engineering standards.
Q: What is a NATO mil?
A: One NATO mil is 1/6400 of a circle, roughly one milliradian. NATO mils are used in artillery and sniping because 1 mil subtends approximately 1 meter at 1000 meters distance.
Q: Can I convert angular velocity with this tool?
A: No — this tool converts static angles only. For angular velocity, convert the angle first, then divide by your time unit separately.
Start using it now
Convert any angle across all seven units instantly with the Angle Converter. For converting other physical quantities, explore the Unit Converter. Work with frequency-to-angular-velocity conversions using the Frequency Converter, or express very large or very small angle values compactly with the Scientific Notation Converter.
