Angle converter for degrees, radians and more
This angle converter switches between seven units: degrees, radians, gradians, arcminutes, arcseconds, full turns and milliradians. It works through degrees internally, using exact relationships wherever they exist.
How the units relate
| Unit | One full turn | In degrees |
|---|---|---|
| Degree (deg) | 360 | 1 |
| Radian (rad) | 2π ≈ 6.283185307 | 180/π ≈ 57.29577951 |
| Gradian (grad) | 400 | 0.9 |
| Arcminute | 21,600 | 1/60 |
| Arcsecond | 1,296,000 | 1/3600 |
| Milliradian (mrad) | 2000π ≈ 6283.185307 | 0.18/π |
| Turn | 1 | 360 |
Degrees and radians
A radian is the angle at which the arc length equals the radius. Since a circle's circumference is 2π radii, a full turn is 2π radians. The two formulas you need are:
radians = degrees × π / 180
degrees = radians × 180 / π
90 deg = 1.570796327 rad (π/2)
180 deg = 3.141592654 rad (π)
1 rad = 57.29577951 degRadians matter in programming because trigonometric functions such as Math.sin() in JavaScript and math.sin() in Python expect them. Passing 90 instead of 1.5708 is a classic bug that returns a strange value instead of 1.
Gradians, turns and CSS
Gradians split a right angle into 100 parts, so 90° = 100 grad. They appear on some surveying equipment and scientific calculators, where a forgotten "GRAD" mode explains many wrong answers. Turns are the most intuitive unit of all: 0.25 turn is a quarter rotation.
CSS accepts deg, rad, grad and turn in transforms and gradients, so rotate(0.25turn), rotate(90deg) and rotate(100grad) are identical.
Arcminutes, arcseconds and milliradians
Astronomy and geography split a degree into 60 arcminutes and each arcminute into 60 arcseconds, so 0.5° = 30 arcmin = 1800 arcsec. The parsec is defined through the arcsecond, which is why the astronomical distance converter is a close relative of this tool.
The milliradian here is the true mathematical one, a thousandth of a radian, giving 6283.185307 per turn. Some military and optics scales use rounded "mils" of 6400 or 6000 per circle; those are different units, so check which one a scope or chart uses.
For rotation rates such as radians per second or rpm, switch to the frequency converter.
Common angles in every unit
These are the values people look up most often. Radians are rounded to six decimals.
| Degrees | Radians | Gradians | Turns |
|---|---|---|---|
| 30° | π/6 ≈ 0.523599 | 33.3333 | 0.083333 |
| 45° | π/4 ≈ 0.785398 | 50 | 0.125 |
| 60° | π/3 ≈ 1.047198 | 66.6667 | 0.166667 |
| 90° | π/2 ≈ 1.570796 | 100 | 0.25 |
| 180° | π ≈ 3.141593 | 200 | 0.5 |
| 360° | 2π ≈ 6.283185 | 400 | 1 |
Two mistakes to avoid
Compass bearings are not math angles. Navigation measures clockwise from north, while math and most graphics APIs measure counterclockwise from the positive x-axis. Convert with math angle = 90° − bearing, so a bearing of 30° is a math angle of 60°.
Normalizing negative angles. In JavaScript and C, -30 % 360 is -30, not 330. Use ((a % 360) + 360) % 360 to bring any angle into the 0–360° range before comparing or displaying it.