The angle units
- Degree (°): a full turn is 360°, a right angle 90°. The everyday unit for angles, compass bearings and geometry.
- Radian (rad): the SI unit. An angle of 1 radian cuts an arc as long as the radius, so a full turn is 2π radians and 1 rad is 180/π, about 57.2958°. Math libraries and physics formulas expect radians.
- Milliradian (mrad): a thousandth of a radian, about 0.0573°. At a distance of 100 m, 1 mrad spans about 10 cm (100 m × 0.001).
- Gradian or gon: a full turn is 400 gon, so a right angle is exactly 100 gon and 1 gon is 0.9°.
- Minute (′) and second (″) of arc: 1° = 60′ and 1′ = 60″, so a degree has 3,600 seconds. Coordinates such as 48° 51′ 24″ N use them.
- Turn: one full revolution, 360°, 2π rad or 400 gon.
The formulas
radians = degrees × π ÷ 180 and degrees = radians × 180 ÷ π. Every other unit is a fixed multiple of the degree, so the converter turns your value into degrees first and then into the unit you want. Negative angles work too: −90° is −π/2. For sines, cosines and other trig functions of an angle, use the scientific calculator. To turn a roof pitch such as 6/12 into degrees, use the roof pitch calculator.
The exact factors, as listed in NIST Special Publication 811, Appendix B:
| Unit | In degrees | In radians |
|---|---|---|
| 1 degree | 1 | π/180 = 0.017 453 29 |
| 1 minute of arc | 1/60 | 0.000 290 888 2 |
| 1 second of arc | 1/3,600 | 0.000 004 848 137 |
| 1 gon (gradian) | 0.9 | π/200 = 0.015 707 96 |
| 1 milliradian | 0.057 295 78 | 0.001 |
| 1 turn (revolution) | 360 | 2π = 6.283 185 |
Worked examples
- 135° to radians: 135 × π ÷ 180 = 3π/4, about 2.3562 rad.
- 2.5 rad to degrees: 2.5 × 180 ÷ π, about 143.2394°.
- 30° to gon: 30 ÷ 0.9 = 33.333 gon. The reverse, gon to degrees, is × 0.9.
- 0.5° to minutes of arc: 0.5 × 60 = 30′. And 0.25′ is 0.25 × 60 = 15″.
- 12° 30′ 36″ to a decimal: 12 + 30 ÷ 60 + 36 ÷ 3,600 = 12.51°. Back again: 0.51 × 60 = 30.6′, and 0.6′ × 60 = 36″.
- 0.75 turn to radians: 0.75 × 2π = 3π/2, about 4.7124 rad, the same as 270°.
Common angles in every unit
| Degrees | Radians, exact | Radians | Gon | Turns |
|---|---|---|---|---|
| 30° | π/6 | 0.5236 | 33.333 | 1/12 |
| 45° | π/4 | 0.7854 | 50 | 1/8 |
| 60° | π/3 | 1.0472 | 66.667 | 1/6 |
| 90° | π/2 | 1.5708 | 100 | 1/4 |
| 120° | 2π/3 | 2.0944 | 133.333 | 1/3 |
| 180° | π | 3.1416 | 200 | 1/2 |
| 270° | 3π/2 | 4.7124 | 300 | 3/4 |
| 360° | 2π | 6.2832 | 400 | 1 |
Angles that are simple fractions of a turn come out as simple fractions of π in radians, which is why math texts write them that way. A value such as 1.5 rad has no neat fraction and is 85.94°.
| From | Equals |
|---|---|
| 1 degree | 0.01745329252 radians |
| 1 radian | 57.29577951 degrees |
| 1 degree | 1.111111111 gradians |
| 1 degree | 60 minutes of arc |
| 1 turn | 6.283185307 radians |
| 1 milliradian | 0.05729577951 degrees |
Degrees to radians
| Degrees (°) | Radians (rad) |
|---|---|
| 1 | 0.0174533 |
| 15 | 0.261799 |
| 30 | 0.523599 |
| 45 | 0.785398 |
| 60 | 1.0472 |
| 90 | 1.5708 |
| 120 | 2.0944 |
| 135 | 2.35619 |
| 150 | 2.61799 |
| 180 | 3.14159 |
| 270 | 4.71239 |
| 360 | 6.28319 |
How to use it
- 1Pick the unit you have on the left and the one you want on the right. Degrees to radians is set up when the page opens.
- 2Type the angle. Either side accepts input, and swap turns the conversion around.
- 3Read the result and see the same angle in every other unit in the list. Copy the number or a link.
Sources
- NIST Special Publication 811, Appendix B.9: factors for degree, minute, second, gon, revolution and mil to radian
- NIST Special Publication 811, Chapter 4: the radian as the SI derived unit of plane angle
- MDN: Math.sin(): takes an angle in radians
Frequently asked questions
How do I convert degrees to radians?
Multiply the degrees by π and divide by 180. 90° is 90 × π ÷ 180 = π/2, about 1.5708 radians, and 45° is π/4, about 0.7854. The other way, multiply radians by 180 and divide by π: 2.5 rad is 2.5 × 180 ÷ π, about 143.24°.
How many degrees is 1 radian?
180 ÷ π, about 57.2958 degrees, or 57° 17′ 45″. A radian is the angle at which the arc of a circle is as long as its radius, which is why a full turn, with a circumference of 2π radii, is 2π radians.
Why do programming languages use radians?
The trigonometric functions in most languages expect radians. In JavaScript, for example, Math.sin takes a number in radians, as the MDN reference states. Convert first: Math.sin(30 * Math.PI / 180) gives the sine of 30°, which is 0.5.
What is a gradian?
A gradian, or gon, divides a right angle into 100 parts, so a full turn has 400 gradians and one gradian is 0.9 degrees. A right angle is a round 100, so tenths and hundredths of a gon stay decimal. 50 gon is 45°.
How many minutes of arc are in a degree?
Sixty. Each minute of arc has 60 seconds of arc, so a degree has 3,600 seconds. To turn 12° 30′ 36″ into a decimal, divide the minutes by 60 and the seconds by 3,600: 12 + 30/60 + 36/3600 = 12.51°.
Is a milliradian the same as a mil?
Not quite. A milliradian is exactly 1/1000 of a radian, so a full turn has 2π × 1000, about 6,283 of them. The military mil in NIST Special Publication 811 is 1/6400 of a turn, about 0.98 mrad. This converter uses the true milliradian.