What Is an Angle Converter?
An angle converter translates angular measurements between degrees, radians, gradians, arcminutes, arcseconds, and revolutions. These units all describe rotation, but they come from different traditions. Degrees date back to ancient Babylonian astronomy, which divided a circle into 360 parts. Radians are the natural mathematical unit, where a full circle is 2 times pi. Gradians were invented for the metric system and divide a circle into 400 parts. Arcminutes and arcseconds subdivide degrees for astronomy, surveying, and optics, where tiny angles matter.
You need angle conversion when a calculus problem gives an angle in radians but your calculator is in degree mode, when a telescope spec lists resolution in arcseconds and you want degrees, or when surveying software uses gradians but a construction drawing uses degrees. For related geometry, see our Triangle Calculator and Law of Cosines Calculator.
What This Calculator Does
This converter handles seven angle units: degrees, radians, milliradians, gradians, arcminutes, arcseconds, and revolutions. Enter a value, pick the source and target units, and the result appears instantly. A swap button reverses the direction, and a full table shows your input in every supported unit at once.
- Inputs: A numeric value, source unit, and target unit
- Outputs: Converted result, conversion details, and a table of all units
How the Conversion Works
The calculator first converts the input to radians, the natural base unit, then divides by the target unit's radian factor. One radian is the angle subtended when the arc length equals the radius. A full circle is 2 times pi radians, or about 6.28318 radians. Since 360 degrees equals 2 pi radians, one degree equals pi divided by 180 radians, or about 0.0174533 radians.
For example, converting 45 degrees to radians: 45 times (pi / 180) equals pi / 4, or about 0.7854 radians. Converting the same 45 degrees to gradians: 45 times (200 / 360) equals 25 gradians, because a half-circle is 180 degrees or 200 gradians.
Common Conversion Factors
| Unit | In Radians | Full Circle |
|---|---|---|
| 1 Degree | 0.0174533 | 360 |
| 1 Radian | 1 | 6.28318 (2 pi) |
| 1 Milliradian | 0.001 | 6,283.18 |
| 1 Gradian | 0.0157080 | 400 |
| 1 Arcminute | 0.000290888 | 21,600 |
| 1 Arcsecond | 0.00000484814 | 1,296,000 |
| 1 Revolution | 6.28318 | 1 |
How to Use the Calculator
- Enter the angle value you want to convert.
- Select the unit you are converting from.
- Select the unit you are converting to, or click swap to reverse the direction.
- Read the result and the full conversion table below it.
Example Calculations
Example 1: Degrees to Radians for Calculus
A student needs to evaluate the sine of 60 degrees, but the textbook formula uses radians.
- Convert: 60 times (pi / 180) = pi / 3 radians
- Decimal: 1.0472 radians
- Check: sin(pi / 3) = 0.8660, which matches sin(60 degrees)
Example 2: Telescope Resolution in Arcseconds to Degrees
A telescope has an angular resolution of 0.5 arcseconds. The user wants this in degrees to compare with a wide-field camera spec.
- Convert: 0.5 arcseconds / 3600 = 0.0001389 degrees
- In arcminutes: 0.5 / 60 = 0.00833 arcminutes
- In radians: 0.5 times 4.84814e-6 = 2.424e-6 radians
Real-World Scenarios
Surveying with Gradians and Degrees
Elena, a land surveyor in France, uses a theodolite that measures in gradians (gon), the standard in much of European surveying. A construction blueprint from a US firm specifies a corner angle of 37.5 degrees. She converts 37.5 degrees to gradians: 37.5 times (200 / 360) equals 41.667 gradians. She sets her instrument to 41.667 gon and verifies the layout. The 400-gradian system makes right angles clean (100 gon each), which is why it was adopted for surveying, but it creates friction whenever European surveyors work with degree-based plans. For distance and area work alongside angles, she uses our Area Calculator.
Long-Range Shooting with Milliradians
Marcus, a competitive long-range shooter, adjusts his rifle scope in milliradians (mrad). His target is at 800 yards, and the wind calls for a 2.5 mrad horizontal adjustment. He wants to understand this in degrees to compare with an older scope that uses arcminutes (MOA, or minutes of angle). Using the converter, 2.5 mrad equals 0.0025 radians, which is 0.1432 degrees, or 8.59 arcminutes. At 800 yards, 1 MOA is about 8.38 inches, so his 8.59 MOA adjustment moves the point of impact about 72 inches, or 6 feet, which matches his wind drift chart. The converter lets him move between the mrad and MOA systems that coexist in the shooting world. For bullet energy calculations, he could pair this with our Kinetic Energy Calculator.
Astronomy and Small Angle Approximation
Priya, an amateur astronomer, reads that the Andromeda Galaxy subtends about 3 degrees by 1 degree on the sky. Her telescope eyepiece has a true field of view of 0.8 degrees, or 48 arcminutes. She converts 3 degrees to arcminutes: 3 times 60 equals 180 arcminutes. Andromeda is far too large for her eyepiece, which only captures 48 arcminutes. She needs binoculars or a short focal length refractor with a wider field. She also checks the angular separation of a double star listed as 4.3 arcseconds: that is 4.3 / 3600 = 0.00119 degrees, or 0.0208 arcminutes, well below her telescope's 1.2 arcsecond resolution limit, so the pair should split cleanly. For light wavelength work, she references our Wavelength Calculator.
Common Mistakes to Avoid
- Calculator in wrong mode: The most common angle error is computing sin, cos, or tan with the calculator in degrees when the input is radians, or vice versa. sin(30) in radian mode is 0.988, not 0.5. Always check the DEG/RAD indicator before evaluating trig functions.
- Confusing arcminutes and arcseconds: One arcminute is 1/60 of a degree. One arcsecond is 1/3600 of a degree, or 1/60 of an arcminute. Mixing them up introduces a 60-fold error. In astronomy and optics, this is the difference between a resolved and unresolved object.
- Treating gradians like degrees: A right angle is 90 degrees but 100 gradians. A full circle is 360 degrees but 400 gradians. If you plug a gradian value into a degree-based formula, every angle is off by about 11%. European surveying instruments use gradians, so check the unit before calculating.
- Forgetting the small angle approximation breaks down: For small angles, sin(x) is approximately x in radians. This is exact only for small x. At 0.1 radians (5.7 degrees), the error is about 0.17%. At 0.5 radians (28.6 degrees), the error is 4.3%. The approximation is a shortcut, not a substitute for the actual sine at larger angles.
Authoritative Research and Resources
- NIST Metric SI Units - The National Institute of Standards and Technology defines the radian as the SI derived unit for angle (a dimensionless unit equal to the ratio of arc length to radius) and provides guidance on the use of degree, radian, and steradian in scientific measurement.
- BIPM - SI Base Units - The International Bureau of Weights and Measures maintains the International System of Units, including the classification of the radian as a dimensionless derived unit and the relationship between radians and degrees in the SI Brochure.