Angle Converter
Convert between degrees, radians, gradians, arcminutes, arcseconds, and revolutions.
How angle conversion works
Every angle unit here is converted through radians as a common base — the natural unit angles are measured in for most math and physics formulas. A full circle is 360 degrees, 2π radians, or 400 gradians, so each unit is just a different way of dividing up the same rotation.
The formulas: radians = degrees × π ÷ 180, and degrees = radians × 180 ÷ π. Gradians are even simpler, since 90 degrees is 100 gradians: gradians = degrees × 10 ÷ 9.
Worked example
Convert 135° to radians and gradians. In radians: 135 × π ÷ 180 = 2.356194, which is 3π ÷ 4. In gradians: 135 × 10 ÷ 9 = 150.
Common conversions
| Conversion | Result |
|---|---|
| 30° in radians | 0.523599 rad |
| 45° in radians | 0.785398 rad |
| 60° in radians | 1.047198 rad |
| 90° in gradians | 100 gon |
| 1 radian in degrees | 57.29578° |
| 1 gradian in degrees | 0.9° |
| 1 arcminute in degrees | 0.016667° |
| 1 degree in arcseconds | 3,600″ |
| 1 revolution in radians | 6.283185 rad |
Choosing the right unit
Degrees are the everyday choice for navigation, construction and design. Radians are what calculus and most programming languages expect, so the sine and cosine functions in a spreadsheet or code usually take radians. Gradians suit surveying, where a right angle splits neatly into 100 parts, and arcminutes and arcseconds describe the small angles used in astronomy and optics.
Frequently asked questions
What are gradians used for?
Gradians (also called gons) divide a right angle into 100 units instead of 90, making a full circle 400 gradians. They're mostly used in surveying and some European engineering contexts, rather than everyday math.
Why do arcminutes and arcseconds matter?
They're used for measuring very small angles precisely — common in astronomy, navigation, and optics, where a single degree is too coarse a unit.
Why does my calculator give the wrong sine or cosine?
It is probably in the wrong angle mode. The sine of 90 is exactly 1 in degree mode, but in radian mode the calculator treats 90 as 90 radians and returns about 0.894.