Angle
Degrees, radians, gradians
About This Calculator
Angles can be measured in degrees (most familiar), radians (used in mathematics and physics), or gradians (used in surveying). A full circle contains 360 degrees, 2π radians, or 400 gradians.
Formula
radians = degrees × (pi / 180)
degrees = radians × (180 / pi)
gradians = degrees × (10 / 9)
1 full circle = 360 deg = 2pi rad = 400 grad
Example Calculation
Convert 90 degrees to radians and gradians.
- radians = 90 × (pi / 180) = pi/2 = 1.5708 rad
- gradians = 90 × (10/9) = 100 grad
90 degrees = 1.5708 radians = 100 gradians
Common Angles Reference
| Degrees | Radians | Gradians |
|---|---|---|
| 0 | 0 | 0 |
| 30 | pi/6 (0.5236) | 33.33 |
| 45 | pi/4 (0.7854) | 50 |
| 60 | pi/3 (1.0472) | 66.67 |
| 90 | pi/2 (1.5708) | 100 |
| 180 | pi (3.1416) | 200 |
| 270 | 3pi/2 (4.7124) | 300 |
| 360 | 2pi (6.2832) | 400 |
Frequently Asked Questions
Why do mathematicians prefer radians over degrees?
Radians make calculus formulas simpler and cleaner. For example, the derivative of sin(x) is cos(x) only when x is in radians. Using degrees introduces a conversion factor into the derivative.
What are gradians used for?
Gradians (also called gon or grad) are used mainly in surveying and civil engineering in some European countries. A right angle is exactly 100 gradians, making it easy to work with decimal fractions of a right angle.
How do I convert radians to degrees quickly?
Multiply by 180/pi (approximately 57.296). So 1 radian is about 57.3 degrees. For common fractions of pi: pi/2 = 90 deg, pi = 180 deg, 2pi = 360 deg.