Convert Turns to Radians
Turns to radians is the conversion you need when you count full rotations but the math you're doing works in radians — trigonometry, physics, or programming. A turn is one whole revolution, and one turn is approximately 6.283185307179586 radians, which is exactly 2π.
Turns to Radians conversion table
| Turns (turn) | Radians (rad) |
|---|---|
| 1 | 6.283185 |
| 2 | 12.566371 |
| 3 | 18.849556 |
| 5 | 31.415927 |
| 10 | 62.831853 |
| 20 | 125.663706 |
| 50 | 314.159265 |
| 100 | 628.318531 |
| 500 | 3,141.592654 |
| 1,000 | 6,283.185307 |
The formula
To convert turns to radians, multiply the number of turns by 6.283185307179586 (which is exactly 2π):
radians = turns × 6.283185307179586
A worked example
Say you have one full turn and want it in radians:
1 × 6.283185307179586 = 6.2831853 rad
So 1 turn is about 6.2831853 radians.
Why it's an approximation
A turn is exactly 2π radians — that part is precise. The catch is π itself: it's an irrational number, so its decimal runs on forever without repeating. That means 2π can never be written out fully as a decimal, and the value 6.283185307179586 is rounded off at some point. It's exact as the expression 2π, but approximate the moment you write it as a decimal.
When you'll use it
Feeding a rotation into trigonometry, physics formulas, or code, where angles are expected in radians rather than full turns or degrees.
Frequently asked questions
How many radians are in a turn?
Approximately 6.283185307179586 radians, which is exactly 2π. Because π never ends, the decimal is rounded — 6.283185307179586 is the practical number to use.
Why is a turn 2π radians?
A radian is defined so that a full circle measures 2π radians, and a turn is one full circle. So one turn equals 2π radians — about 6.283185307179586.
Is the turns to radians conversion exact?
No. A turn is exactly 2π radians, but π is irrational — it has no finite decimal — so any decimal like 6.283185307179586 is rounded. More decimal places give you more precision.