Angle Converter
Convert angles between degrees, radians, gradians, arcminutes, arcseconds, milliradians, NATO mils and turns.
Updated Formula covered by automated tests
Angles turn up in more units than most people expect: degrees on a protractor or compass, radians in every calculus and programming library, gradians (gon) on survey instruments, arcminutes and arcseconds in astronomy and map coordinates, and milliradians or minutes of angle on rifle and telescope sights. Enter an angle in any of them and this converter shows the equivalent in all the rest.
A common reason to use it is a mismatch between a tool and its documentation, for example a spreadsheet trigonometry function that expects radians, a total station set to gon, or a scope that adjusts in 0.1 mrad clicks while a ballistic table lists minutes of angle. Negative angles are accepted, so clockwise and counterclockwise rotations both convert correctly.
The formula
- °
- Degree, 1/360 of a full turn
- rad
- Radian, the angle whose arc length equals the radius of the circle
- gon
- Gradian, 1/400 of a turn, so a right angle is exactly 100 gon
- ′ and ″
- Arcminute (1/60 degree) and arcsecond (1/60 arcminute)
- π
- Pi, the ratio of a circle’s circumference to its diameter, about 3.14159
The 360-degree circle and its division into sixtieths come from Babylonian astronomy, which used base-60 arithmetic; 360 also divides evenly by 2, 3, 4, 5, 6, 8, 9, 10 and 12, which made it convenient for hand calculation. The radian is different in kind: it is defined by geometry rather than by convention, as arc length divided by radius. That is why the SI treats it as its coherent unit of plane angle and why formulas such as arc length = r × θ or the derivative of sin x being cos x only hold when θ and x are in radians.
The gradian came out of the French metric reforms in the 1790s, giving a right angle of 100 units, and it survives in surveying in parts of Europe. Military mils approximate the milliradian so that one unit subtends about one meter at a distance of one kilometer; NATO rounded the count to 6400 per circle because it divides evenly by two many times, while a true circle holds about 6283.19 milliradians.
Worked example
Dana’s ballistic app says her rifle needs 2.5 MOA of elevation to hit at a new distance, but her scope turrets click in milliradians. She wants the correction in mrad and the shift it represents at 100 m.
- 1 Convert minutes of angle to degrees: 2.5 ÷ 60 = 0.041667°.
- 2 Convert degrees to milliradians: 0.041667 × π ÷ 180 × 1000 = 0.7272 mrad.
- 3 Turn that into a linear offset at 100 m: 100 m × tan(0.0007272) = 0.0727 m, or about 7.27 cm.
- 4 Round to the nearest click: with 0.1 mrad clicks that is 7 clicks (0.7 mrad).
Result: 2.5 MOA is 0.727 mrad, about 7.3 cm at 100 m, so seven 0.1 mrad clicks get within about 0.3 cm of the requested correction at that range.
Before you rely on the result
- Results are pure angle conversions. Turning an angle into a distance (as in the scope example) uses trigonometry that assumes a flat target perpendicular to the line of sight.
- Only the NATO 6400 mil is included; if your equipment uses another mil system, convert through degrees using its own count per circle.
Frequently asked questions
Why does my calculator give a strange answer for sin(90)?
It is probably in radian mode. sin(90 rad) is about 0.894, while sin(90°) is exactly 1. Spreadsheets and programming languages almost always expect radians, so convert first with degrees × π ÷ 180, or use a built-in function such as RADIANS() in Excel.
Is 1 MOA really one inch at 100 yards?
Close but not exact. One minute of angle subtends 1.047 inches at 100 yards, or 2.91 cm at 100 m. Some shooters use "shooter’s MOA" of exactly 1 inch per 100 yards, which is about 4.5 percent smaller; check which one a scope uses.
How do I convert GPS coordinates from degrees-minutes-seconds to decimal degrees?
Add the parts: degrees + minutes ÷ 60 + seconds ÷ 3600. For example 40° 26′ 46″ is 40 + 0.43333 + 0.01278 = 40.44611°. Southern latitudes and western longitudes become negative in decimal form.
What is the difference between a milliradian and a mil on a compass?
A milliradian is exactly 1/1000 of a radian, giving about 6283 per circle. Military compasses use rounded systems instead: NATO uses 6400 mils per circle, and some other armies have used 6000 or 6300, so the same printed number can mean slightly different angles.