Angle Converter
Angle converter for geometry and trigonometry — degrees, radians, and gradians.
Convert between degrees, radians, gradians, arcminutes and arcseconds. Handy for trigonometry, programming, surveying, navigation and astronomy.
Popular Angle Conversions
- Degree to Radian
- Radian to Degree
- Degree to Gradian
- Arcminute to Degree
- Arcsecond to Degree
- Degree to Arcminute
All Angle Units
| Unit | Symbol | Definition |
|---|---|---|
| Degree | ° | One 360th of a full rotation. |
| Radian | rad | The SI unit of angle, where one full rotation equals 2π radians. |
| Gradian | grad | One 400th of a full rotation, so a right angle is 100 gradians. |
| Arcminute | ' | One 60th of a degree. |
| Arcsecond | " | One 60th of an arcminute, or 1/3600 of a degree. |
How angle conversion works
A full turn is 360 degrees, 2π radians or 400 gradians. The radian is the SI unit: one radian is the angle at the centre of a circle that subtends an arc equal in length to the radius.
degrees = radians × 180 / π
1 rad = 57.29578°
1° = 0.017453293 rad
1° = 60 arcminutes = 3,600 arcseconds
90° = 100 gradians
Common angles in degrees and radians
| Degrees | Radians (exact) | Radians (decimal) | Gradians |
|---|---|---|---|
| 0° | 0 | 0 | 0 grad |
| 30° | π/6 | 0.523599 | 33.333 grad |
| 45° | π/4 | 0.785398 | 50 grad |
| 60° | π/3 | 1.0472 | 66.667 grad |
| 90° | π/2 | 1.5708 | 100 grad |
| 120° | 2π/3 | 2.0944 | 133.33 grad |
| 135° | 3π/4 | 2.35619 | 150 grad |
| 180° | π | 3.14159 | 200 grad |
| 270° | 3π/2 | 4.71239 | 300 grad |
| 360° | 2π | 6.28319 | 400 grad |
Degrees, minutes and seconds (DMS)
Map coordinates and astronomy use degrees (°), arcminutes (′) and arcseconds (″). To convert DMS to decimal degrees: degrees + minutes ÷ 60 + seconds ÷ 3,600. For example, 24° 51′ 36″ = 24 + 0.85 + 0.01 = 24.86°. One arcminute of latitude is one nautical mile, which is how the nautical mile originated.
Where each unit is used
- Degrees: geometry, construction, compass bearings, slopes and everyday angles.
- Radians: calculus, physics, signal processing and nearly all programming languages' trigonometric functions.
- Gradians: some surveying and civil-engineering instruments in continental Europe, where a right angle is 100 grad.
- Arcminutes and arcseconds: astronomy, telescope pointing, geodesy and GPS coordinates.
Common mistakes to avoid
- Calculator or code in the wrong mode. sin(30) is 0.5 in degree mode but −0.988 in radian mode. In JavaScript, Python and Excel, trigonometric functions expect radians, so convert degrees first.
- Mixing arcminutes with time minutes. An arcminute is 1/60 of a degree and has nothing to do with clock minutes.
- Rounding π. Use the full value of π (3.14159265…) rather than 3.14 for precise work. Using 3.14 gives an error of about 0.05%.
- Treating 100 grad as 100°. A right angle is 90° but 100 grad.
Exact conversion factors
| Unit | Symbol | Exact factor |
|---|---|---|
| Degree | ° | 1 ° = 0.0174532925 rad |
| Radian | rad | 1 rad = 1 rad |
| Gradian | grad | 1 grad = 0.0157079633 rad |
| Arcminute | ' | 1 ' = 0.0002908882 rad |
| Arcsecond | " | 1 " = 0.0000048481 rad |
Sources: BIPM, The International System of Units (SI Brochure), on the radian; NIST Special Publication 811.
Frequently Asked Questions
How do I convert degrees to radians?
Multiply degrees by π/180 (about 0.0174533). For example, 90° = 1.5708 rad = π/2.
How do I convert radians to degrees?
Multiply radians by 180/π (about 57.2958). For example, 1 rad = 57.2958°.
How many degrees are in a radian?
1 radian is 57.29578 degrees.
How many arcseconds are in a degree?
There are 3,600 arcseconds in a degree (60 arcminutes × 60 arcseconds).
How many gradians are in a right angle?
A right angle is 100 gradians (grad), since a full circle is 400 grad.
What is 180 degrees in radians?
180° is π radians, or 3.141593 rad.
All Angle Conversions
Choose a starting unit to see every conversion available from it.