This angle converter changes an angle between degrees, radians, gradians, arcminutes, arcseconds and full turns. All values go through the radian, the SI coherent unit: one full turn is 2π rad = 360° = 400 gradians. So 1° = π/180 rad, about 0.0174533 rad, and 1 rad is about 57.2958°.
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How to use the angle converter
- Enter the angle, for example 45.
- In the angle converter, choose its unit and the target unit, such as degrees to radians.
- Read the angle converter result: 45° = 0.785398 rad (π/4).
Calculators and programming languages expect radians in trigonometric functions unless set to degree mode, which is a common source of wrong answers.
What is angles?
An angle measures rotation between two lines. The SI-derived unit is the radian; a full turn is 2π radians or 360 degrees. Degrees, arcminutes and arcseconds dominate navigation and astronomy, gradians appear in surveying, and milliradians are used in optics and ballistics.
Conversion formula
Every unit is stored as a multiple of the base unit, the radian (rad). To convert:
value in target unit = value × (factor of source unit ÷ factor of target unit)
where each factor is the size of the unit in radians.
Common angles conversions
| From | To | 1 unit equals |
|---|---|---|
| degree (°) | radian (rad) | 0.01745329 rad |
| radian (rad) | degree (°) | 57.29578 ° |
| degree (°) | gradian (gon) | 1.111111 gon |
| turn (turn) | degree (°) | 360 ° |
| degree (°) | arcminute (′) | 60 ′ |
| arcminute (′) | arcsecond (″) | 60 ″ |
| gradian (gon) | degree (°) | 0.9 ° |
| milliradian (mrad) | degree (°) | 0.05729578 ° |
Angle converter worked example
Convert 25 degrees to gradians:
- Factor of ° = 0.017453293 rad
- Factor of gon = 0.015707963 rad
- 25 × 0.017453293 ÷ 0.015707963 = 27.77778 gon
Angles unit reference
| Unit | Symbol | Group | Equals (rad) |
|---|---|---|---|
| Radian | rad | SI | 1 |
| Degree | ° | Common | 0.01745329252 |
| Gradian | gon | Metric | 0.01570796327 |
| Arcminute | ′ | Common | 0.0002908882087 |
| Arcsecond | ″ | Common | 4.84814 × 10−6 |
| Turn (revolution) | turn | Common | 6.283185307 |
| Milliradian | mrad | SI | 0.001 |
| Quadrant | quad | Common | 1.570796327 |
| Sextant | sext | Common | 1.047197551 |
Reverse factors for each angles unit
The reference table gives each unit in radians. To go the other way, divide by that factor; the list below does the division for you.
- Degree (°), Common: 1 ° = 0.01745329 rad, so 1 rad = 57.29578 °.
- Gradian (gon), Metric: 1 gon = 0.01570796 rad, so 1 rad = 63.66198 gon.
- Arcminute (′), Common: 1 ′ = 0.0002908882 rad, so 1 rad = 3,437.747 ′.
- Arcsecond (″), Common: 1 ″ = 4.84814 × 10−6 rad, so 1 rad = 206,264.7 ″.
- Turn (revolution) (turn), Common: 1 turn = 6.283185 rad, so 1 rad = 0.1591549 turn.
- Milliradian (mrad), SI: 1 mrad = 0.001 rad, so 1 rad = 1,000 mrad.
- Quadrant (quad), Common: 1 quad = 1.570796 rad, so 1 rad = 0.6366198 quad.
- Sextant (sext), Common: 1 sext = 1.047198 rad, so 1 rad = 0.9549297 sext.
Standard angles in radians
Multiply degrees by π/180 to get radians. For radians to degrees, multiply by 180/π. Standard angles:
| Degrees | Radians (exact) | Radians (decimal) | Gradians |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 30 | π/6 | 0.523599 | 33.3333 |
| 45 | π/4 | 0.785398 | 50 |
| 60 | π/3 | 1.047198 | 66.6667 |
| 90 | π/2 | 1.570796 | 100 |
| 180 | π | 3.141593 | 200 |
| 270 | 3π/2 | 4.712389 | 300 |
| 360 | 2π | 6.283185 | 400 |
A gradian (gon) is 1/400 of a turn, 0.9°. It is used mainly in surveying.
Degrees, minutes and seconds (DMS)
Map coordinates are often written in degrees minutes seconds. One degree is 60 arcminutes and one arcminute is 60 arcseconds. To convert 40.7128° to DMS: the whole part is 40°; 0.7128 × 60 = 42.768, so 42′; 0.768 × 60 = 46.08, so 46.08″. The result is 40° 42′ 46.08″. To go back, use degrees + minutes/60 + seconds/3,600. For triangles, try the right triangle calculator.
Slopes, roof pitch and mils
Many trades state angles as a ratio instead of degrees. A road grade of 5% rises 5 units per 100 horizontal, which is atan(0.05) = 2.86°. A roof pitch of 6 in 12 is atan(0.5) = 26.57°. Convert the ratio to degrees first, then use the angle converter for radians or gradians.
Military and surveying equipment may use mils. NATO uses 6,400 mils in a circle, so 1° is about 17.78 mils; some other systems use 6,000 or 6,283. Gradians split a right angle into 100, so a full circle is 400 gon. The angle converter handles these units, but check which mil system a device uses before relying on the result.
Related converters and calculators
For trigonometry and rotation:
- Degrees to radians: single-pair chart
- Radians to degrees: single-pair chart
- Right triangle calculator: sides and angles
- Triangle calculator: any triangle
- Angular velocity converter: rpm and rad/s
- Scientific calculator: trig functions
Sources
BIPM: SI Brochure; NIST Special Publication 811, Appendix B.
Questions about this calculator
What is the SI unit of angle?
The radian (rad), the angle that subtends an arc equal in length to the radius. A full circle is 2π rad.
How do I convert degrees to radians?
Multiply by π/180. For example, 90° = π/2 ≈ 1.5708 rad.
How do I convert radians to degrees?
Multiply by 180/π. For example, 2 rad ≈ 114.592°.
How do I convert decimal degrees to DMS?
Take the whole degrees, multiply the remainder by 60 for minutes, then multiply that remainder by 60 for seconds. 12.5° = 12° 30′ 0″.
What is a gradian?
A gradian (gon) is 1/400 of a full turn, exactly 0.9°. A right angle is 100 gradians.
How many degrees is one radian?
About 57.2958°, since 1 rad = 180/π degrees.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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