Angle Converter
Convert between degrees, radians, and other angle units.
About the Angle Converter
The Angle Converter turns an angle written in one unit into the same angle written in another — degrees to radians for trigonometry, degrees to gradians for surveying, or degrees to arcminutes and arcseconds for astronomy and navigation. Type a number, pick the two units, and the answer updates as you type. Nothing leaves your browser.
How to convert degrees to radians
A full turn is 360 degrees, and it is also 2π radians. Dividing both by 360 gives the factor: one degree is π/180 radians, which is 0.01745329 rad.
Worked example. A 45° angle is 45 × π/180 = 0.7853982 rad, which is π/4. Going back, 1 rad is 57.29578°, so a 2 rad arc spans 114.5916°.
Degrees to radians conversion table
| Degrees (°) | Radians (rad) |
|---|---|
| 1 | 0.01745329 |
| 2 | 0.03490659 |
| 3 | 0.05235988 |
| 5 | 0.08726646 |
| 10 | 0.1745329 |
| 20 | 0.3490659 |
| 50 | 0.8726646 |
| 100 | 1.745329 |
| 250 | 4.363323 |
| 500 | 8.726646 |
| 1,000 | 17.45329 |
Other exact angle factors
- 1 degree = 60 arcminutes = 3,600 arcseconds, exactly.
- 1 gradian = 0.9 degrees exactly, so a full turn is 400 gradians.
- 1 radian = 57.29578 degrees = 3,437.747 arcminutes.
- A right angle is 90°, π/2 rad, or 100 gradians.
Real-world values
- A full turn: 360°, 2π rad (6.283185 rad), or 400 gradians.
- A right angle: 90°, 1.570796 rad, 100 gradians.
- The Sun and Moon each appear about 0.5° wide from Earth, which is 30 arcminutes.
- A 1-in-12 wheelchair ramp rises at 4.764°, or 0.08314 rad.
- Earth's axial tilt is 23.4393°, written 23° 26′ 21″ in degrees, minutes and seconds.
- Hubble resolves detail near 0.05 arcseconds, which is 1.389 × 10⁻⁵ degrees.
What is a radian?
A radian is the angle you get when the arc you have walked around a circle is exactly as long as the circle's radius. That makes it the natural unit for anything circular: arc length is simply radius times angle in radians, and the calculus of sine and cosine only comes out cleanly in radians. Programming languages follow the same rule — Math.sin in JavaScript, Python and C all expect radians, which is the most common reason people need this conversion.
Degrees, minutes and seconds in practice
Latitude and longitude are still written in degrees, arcminutes and arcseconds, so a coordinate such as 51° 28′ 38″ N has to be converted before a mapping API will accept it. Divide the arcminutes by 60 and the arcseconds by 3,600, then add both to the whole degrees: that coordinate becomes 51.4772°. This converter does the two divisions for you one at a time.
Units this converter supports
Five: degree, radian, gradian, arcminute and arcsecond. Any pair converts in either direction. Degrees, arcminutes and arcseconds form a sexagesimal ladder of sixties; radians and gradians are separate systems tied to the full turn.
Frequently asked questions
How many radians is one degree?
One degree is π/180 radians, which is 0.01745329 rad. A full turn is 360 degrees or 2π radians, so one radian is 180/π degrees, about 57.29578 degrees.
What is a gradian?
A gradian, also written gon or grad, divides a right angle into 100 parts, so a full turn is 400 gradians and one gradian is exactly 0.9 degrees. Surveying and some European civil engineering practice use it because slopes in percent map onto it cleanly.
How do arcminutes and arcseconds relate to degrees?
They subdivide the degree in sixties. One degree is 60 arcminutes, and one arcminute is 60 arcseconds, so one degree is 3600 arcseconds. Astronomy, navigation and optics use them to state very small angles without long decimals.
Which units does this converter support?
Five: degree, radian, gradian, arcminute and arcsecond. Any pair converts in either direction, so the same tool handles degrees to radians for trigonometry and degrees to arcseconds for an astronomical coordinate.
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