How many radians is 360 degrees?
360° equals 2π radians, which is approximately 6.28318 rad.
Angle calculator
Angle converter
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Angle measures rotation or the inclination between two lines. The SI base unit is the radian (rad), defined as the angle subtended at the center of a circle by an arc equal in length to the radius. The common everyday unit is the degree (°): 360° = 2π rad in a full turn. Other units include gradians (400 in a full turn), arcminutes (1/60 of a degree), and arcseconds (1/3,600 of a degree).
Example angle conversions
1 full turn360°, 2π rad, 400 gradians
180°π rad (≈ 3.14159), 200 gradians
90° (right angle)π/2 rad, 100 gradians
60°π/3 rad
Open the live angle converter for any input value.
Radians vs degrees
Radians are the natural mathematical unit for angle. They make trigonometric and calculus identities clean: the derivative of sin(x) is cos(x) only when x is in radians, and Taylor series for sin and cos converge directly without conversion factors. This is why physics and engineering default to radians.
Degrees are the everyday practical unit. They divide a full turn into 360 parts, a number chosen historically because 360 has many divisors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180), making it convenient for angle arithmetic with whole numbers. The Babylonian preference for base-60 likely fixed the convention.
Arcminutes and arcseconds
For fine angular measurement, degrees subdivide into arcminutes (1/60 of a degree) and arcseconds (1/60 of an arcminute, or 1/3,600 of a degree). The notation uses single and double primes: 12° 34' 56" is 12 degrees, 34 arcminutes, 56 arcseconds.
Astronomy uses arcseconds heavily because celestial objects are measured to fractions of an arcsecond. The Hubble Space Telescope's angular resolution is about 0.05 arcseconds. 1 arcsecond at a distance of 1 parsec is the definition of 1 astronomical unit, which is the basis for the parsec unit itself (about 3.26 light-years).
Gradians and other niche units
Gradians (or 'gons') divide a full turn into 400 parts, so a right angle is 100 grad. They were invented during the French Revolution as part of the broader metrication push but never gained traction outside specialized surveying and some European mathematics curricula. Most calculators offer a GRAD mode alongside DEG and RAD.
Other rarely-seen units: turn (1 full rotation = 360°), revolution (often used interchangeably with turn), and binary degrees in some programming contexts where 256 or 65,536 units divide the circle for efficient integer arithmetic.
Common angle conversions
| From | Equivalent |
|---|---|
| 1 full turn | 360°, 2π rad, 400 gradians |
| 180° | π rad (≈ 3.14159), 200 gradians |
| 90° (right angle) | π/2 rad, 100 gradians |
| 60° | π/3 rad |
| 45° | π/4 rad |
| 30° | π/6 rad |
| 1 radian | 57.2958°, 63.6620 gradians |
| 1 degree | 0.0174533 rad, 60 arcminutes |
| 1 arcminute | 1/60°, 0.000291 rad |
| 1 arcsecond | 1/3,600°, 4.848 × 10⁻⁶ rad |
Frequently asked questions
Why do mathematicians prefer radians?
Trig and calculus identities are clean in radians. The derivative of sin(x) is cos(x) and Taylor series converge correctly only when x is in radians. Using degrees introduces extra conversion factors.
What's an arcsecond?
1/3,600 of a degree, used heavily in astronomy and surveying for fine angular measurement. 1 arcsecond at a distance of 1 parsec defines an astronomical unit.
What are gradians and why are they uncommon?
Gradians divide a full turn into 400 parts (100 per right angle). They were a French Revolution metrication experiment that survived in some surveying applications but never displaced degrees.
How do I convert between degrees and radians?
radians = degrees × π/180. degrees = radians × 180/π. For quick checks: 1 rad is about 57.3°, π/2 rad is 90°, π rad is 180°.