The units of angle
Degrees are a convention inherited from Babylonian astronomy, which divided the circle into 360 parts because 360 has so many divisors. Radians are the natural unit: one radian is the angle where the arc equals the radius, so a full circle is 2π radians. Every trigonometric identity in calculus assumes radians, which is why derivatives of sine and cosine come out clean only in that unit.
| Unit | Symbol | Equals |
|---|---|---|
| Radians | rad | 1 rad |
| Degrees | ° | 0.01745329 rad |
| Gradians | gon | 0.01570796 rad |
| Turns | turn | 6.28318531 rad |
| Arcminutes | ' | 0.00029089 rad |
| Arcseconds | " | 0.00000485 rad |
How to convert by hand
Every angle conversion follows the same procedure: take the amount to the base unit, then from there to the target unit. In practice it is a single multiplication, because the two steps combine into one factor.
Multiply degrees by π/180 for radians, or radians by 180/π for degrees. A full turn is 360 degrees, 2π radians, 400 gradians or 1 turn.
The equivalents people look up most
| Conversion | Result |
|---|---|
| A right angle | 90° · π/2 rad · 100 gon |
| A full circle | 360° · 6.283 rad · 400 gon |
| A degree | 60 arcminutes · 3,600 arcseconds |
| 1 radian | 57.296° |
| A half turn | 180° · π rad |
Common mistakes
- Leaving a calculator in the wrong mode. sin(30) is 0.5 in degrees and −0.988 in radians, and nothing warns you.
- Using degrees in a calculus formula. The derivative of sin(x) is cos(x) only when x is in radians.
- Confusing gradians with degrees. A gradian is 0.9 degrees, so 100 gon is a right angle, not 100°.
- Writing an arcminute as a prime and a minute of time the same way. They are different quantities that share a symbol.