Angle Unit Converter
Angles can be expressed in several units, each suited to different contexts. Degrees (360 per full circle) are universal in everyday use, navigation, and engineering. Radians (2*pi per circle) are standard in mathematics and physics. Gradians (400 per circle) appear in surveying. Arcminutes and arcseconds provide high-resolution subdivision of degrees for astronomy and geodesy. This converter handles all five units simultaneously using exact relationships derived from the definition of a full circle.
Angle conversion factors
1 turn = 360 deg = 2*pi rad = 400 grad = 21,600 arcmin = 1,296,000 arcsec
All conversions route through degrees. The radian is the SI derived unit. 1 rad = 180/pi degrees exactly. 1 grad = 9/10 degree exactly. 1 arcmin = 1/60 degree exactly. 1 arcsec = 1/3600 degree exactly.
Angle unit applications
- Degrees: Navigation, engineering drawing, common usage.
- Radians: Mathematics, physics, programming (trigonometric functions).
- Gradians: European surveying and civil engineering.
- Arcminutes: Navigation, GPS (1 arcmin latitude = 1 nautical mile).
- Arcseconds: Astronomy (stellar positions), high-precision geodesy.
Angle unit converter: frequently asked questions
How do I convert degrees to radians?
Multiply degrees by pi/180. For example, 90 degrees = 90 * pi/180 = pi/2 = 1.5708 radians. Alternatively, 180 degrees = pi radians, and 360 degrees = 2*pi radians.
What is a gradian (grad)?
A gradian (also called grade or gon) divides a right angle into 100 parts, so a full circle is 400 gradians. Gradians are used primarily in surveying and civil engineering in some European countries. One degree = 10/9 gradians. A right angle is exactly 100 gradians.
What is an arcminute and arcsecond?
An arcminute is 1/60 of a degree (written with the symbol '). An arcsecond is 1/60 of an arcminute, or 1/3600 of a degree (written with the symbol ''). These units are used in astronomy and GPS positioning. Earth's circumference spans about 1 nautical mile per arcminute of latitude.
Why do mathematicians use radians instead of degrees?
Radians simplify formulas in calculus and trigonometry. The derivative of sin(x) is cos(x) only when x is in radians. Arc length formula is simply s = r*theta in radians. Degrees introduce constant factors of pi/180 everywhere in calculus.
How many degrees is one radian?
One radian equals 180/pi degrees, approximately 57.2958 degrees. This comes from the definition: an angle of 1 radian subtends an arc length equal to the radius on a circle.
Official sources
- NIST Special Publication 811: Guide for the Use of the International System of Units (SI).
- BIPM SI Brochure: The radian as SI supplementary unit.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.