Everything anchors to the half turn
Convert degrees to radians by multiplying by π/180; go the other way with 180/π (one radian ≈ 57.3°). Gradians use 400 to the turn, so a right angle is a clean 100 gon.
Common angles across every unit
| Degrees | Radians | Gradians | Turns |
|---|---|---|---|
| 30° | π/6 ≈ 0.5236 | 33.33 gon | 1/12 |
| 45° | π/4 ≈ 0.7854 | 50 gon | 1/8 |
| 60° | π/3 ≈ 1.0472 | 66.67 gon | 1/6 |
| 90° | π/2 ≈ 1.5708 | 100 gon | 1/4 |
| 180° | π ≈ 3.1416 | 200 gon | 1/2 |
| 360° | 2π ≈ 6.2832 | 400 gon | 1 |
One degree also splits into 60 arcminutes (′) and 3,600 arcseconds (″) for fine astronomy and navigation work.
The mistake that ruins trig answers
A calculator or programming function reads an angle in whichever mode it is set to. Feed it the wrong unit and the result is silently, badly wrong — not an error, just a different number.
Same keystrokes, two answers. Trig functions in code always take radians, so convert degrees with × π/180 before calling sin, cos or tan.
Common questions
How many radians is 90 degrees?
90 degrees is pi/2 radians, about 1.5708. To go from degrees to radians, multiply by pi/180.
What is the difference between degrees and radians?
Degrees split a full turn into 360 parts, a historical choice. Radians split it into 2 pi parts, which falls out of circle geometry: one radian is the angle where the arc equals the radius. So 180 degrees equals pi radians. Calculus formulas assume radians.
What is a gradian?
A gradian, or gon, divides a full turn into 400 parts, so a right angle is exactly 100 gon and 1 gon is 0.9 degrees. It comes from the metric era and survives mainly in European surveying.
