Angle conversion formulas
| From | To | Formula |
|---|---|---|
| Degrees | Radians | rad = deg × π / 180 |
| Radians | Degrees | deg = rad × 180 / π |
| Degrees | Gradians | grad = deg × 10 / 9 |
| Degrees | Turns | turns = deg / 360 |
| Decimal degrees | DMS | D = whole degrees; M = ⌊(deg − D) × 60⌋; S = remainder × 60 |
| DMS | Decimal degrees | deg = D + M/60 + S/3600 |
| Degrees | Milliradians | mrad = deg × 1000π / 180 ≈ deg × 17.4533 |
Common angles in every unit
| Degrees | Radians (exact) | Radians | Gradians | DMS |
|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 0° 0′ 0″ |
| 1° | π/180 | 0.0175 | 1.1111 | 1° 0′ 0″ |
| 15° | π/12 | 0.2618 | 16.6667 | 15° 0′ 0″ |
| 30° | π/6 | 0.5236 | 33.3333 | 30° 0′ 0″ |
| 45° | π/4 | 0.7854 | 50 | 45° 0′ 0″ |
| 60° | π/3 | 1.0472 | 66.6667 | 60° 0′ 0″ |
| 90° | π/2 | 1.5708 | 100 | 90° 0′ 0″ |
| 120° | 2π/3 | 2.0944 | 133.3333 | 120° 0′ 0″ |
| 135° | 3π/4 | 2.3562 | 150 | 135° 0′ 0″ |
| 150° | 5π/6 | 2.618 | 166.6667 | 150° 0′ 0″ |
| 180° | π | 3.1416 | 200 | 180° 0′ 0″ |
| 225° | 5π/4 | 3.927 | 250 | 225° 0′ 0″ |
| 270° | 3π/2 | 4.7124 | 300 | 270° 0′ 0″ |
| 315° | 7π/4 | 5.4978 | 350 | 315° 0′ 0″ |
| 360° | 2π | 6.2832 | 400 | 360° 0′ 0″ |
Which angle unit should I use?
- Degrees are what protractors, miter saws, roof pitch gauges and most everyday tools use. A full circle is 360°.
- Radians are the natural unit in math and physics: calculus, trigonometric functions in code (
Math.sinexpects radians) and rotational motion. A full circle is 2π ≈ 6.2832 rad. - Gradians split a right angle into 100 parts. You will find them on some scientific calculators (the GRAD mode) and surveying instruments.
- DMS is the traditional way to write latitude, longitude and survey bearings, e.g. 40° 44′ 55″ N.
- MOA and milliradians are used for rifle scopes and rangefinding. 1 MOA is about 1.047 inches at 100 yards.
Need to see an angle rather than convert it? Try the online protractor or browse the angle library.
Frequently asked questions
How do I convert degrees to radians?
Multiply by π and divide by 180: radians = degrees × π / 180. For example 45° × π/180 = π/4 ≈ 0.7854 rad. See the full degrees to radians guide.
How do I convert radians to degrees?
Multiply by 180 and divide by π: degrees = radians × 180 / π. One radian is about 57.2958°.
What is a gradian?
A gradian (also called a gon or grad) is 1/400 of a full turn, so a right angle is exactly 100 gradians. It is used in some surveying and European engineering work. 1° = 10/9 grad ≈ 1.1111 grad.
What does DMS mean?
DMS stands for degrees, minutes, seconds. One degree is split into 60 minutes (′), and each minute into 60 seconds (″). So 47.5° = 47° 30′ 0″. DMS is used for latitude and longitude, surveying and astronomy.
How many milliradians (mils) are in a degree?
One degree is about 17.4533 milliradians (mrad). Shooters and rangefinder users often work in mrad or in minutes of angle (MOA), where 1 MOA = 1/60° ≈ 0.2909 mrad.