Turns to Radians
Type a value in either box to convert between turns and radians. The factor is 6.28319.
| Turns | Radians |
|---|---|
| 1 turn | 6.283185 rad |
| 2 turn | 12.566371 rad |
| 3 turn | 18.849556 rad |
| 5 turn | 31.415927 rad |
| 10 turn | 62.831853 rad |
| 25 turn | 157.079633 rad |
| 50 turn | 314.159265 rad |
| 100 turn | 628.318531 rad |
How to use
- Enter a value in the Turns field.
- The Radians figure appears immediately — no button to press.
- Typing in the Radians field converts back to Turns the same way.
- Use the reference table for the values that come up most often.
Turns to Radians formula
Converting turns to radians is a single multiplication by 6.28318531.
rad = turn × 6.28318531
Why this factor: the turn and the radian are both defined against the degree, so dividing one definition by the other gives 6.28318531 directly. Both definitions are set out in full below.
To go the other way, divide by 6.28318531 — or use the Radians to Turns converter, which is the same calculation reversed.
Worked example
Start with 30 turn. Multiply by 6.28318531 to get 188.495559 rad.
30 turn × 6.28318531 = 188.495559 rad
The converter above does this for any value you type, in either direction.
What is a turn?
A turn is one complete revolution — a full circle.
Definition: Exactly 360 degrees, or 2π radians.
Where it is used: Rotational counting, gear and pulley ratios, and coil and winding specifications.
What is a radian?
A radian is the angle that subtends an arc equal in length to the circle's radius — the natural unit of angle.
Definition: A full turn is exactly 2π radians, so one radian is 180/π degrees, about 57.29578°.
Where it is used: Calculus and trigonometry, physics, and every standard programming library's sin, cos and tan.
Worth knowing: Programming maths functions almost always expect radians. Passing degrees to Math.sin() is one of the most common numerical bugs.
Turns vs Radians
The turn is a specialized unit while the radian is an SI unit. That is why the factor between them is not a round number: the two systems were defined independently and only later tied together by exact agreement.
Frequently asked questions
How many radians are in a turn?
One turn equals 6.28318531 radians.
What is the formula to convert turns to radians?
Multiply the number of turns by 6.28318531. To reverse it, divide by 6.28318531 instead.
How do I convert radians back to turns?
Divide by 6.28318531, which is the same as multiplying by 0.159154943.
How many radians are in 10 turns?
10 turn works out to 62.831853 rad.
Is there anything to watch out for with radians?
Programming maths functions almost always expect radians. Passing degrees to Math.sin() is one of the most common numerical bugs.
How is the turn defined?
Exactly 360 degrees, or 2π radians.