Convert Turns to Radians
Convert Turns (rev) to Radians (rad) instantly and accurately.
Conversion Formula
rad = rev × 6.283185307
About Turns
A turn (revolution, cycle) equals one complete rotation - 2π rad = 360° = 400 grad. Rotational speed is naturally measured in turns per minute (rpm): a car engine idles at ~750 rpm; a hard disk spins at 7,200 rpm. The factor 2π in physics formulas (ω = 2πf, E = hf = ħω) arises from converting turns to radians. The turn is the natural unit for gear ratios, transformer winding counts, and thread pitch (turns per inch). The tau convention (τ = 2π ≈ 6.2832) proposes 1τ = 1 full turn. 1 rpm = 1/60 Hz = 2π/60 rad/s ≈ 0.10472 rad/s.
About Radians
The radian (rad) is the SI unit of plane angle, defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. A full circle = 2π rad; a right angle = π/2 rad. The radian is dimensionless (arc length / radius), so it can be omitted in equations. All calculus and physics formulas are simplest in radians: d/dx(sin x) = cos x holds only in radians; Taylor series sin x = x − x³/6 + x⁵/120 hold only for x in radians; angular frequency ω = 2πf gives energy E = ħω. 1 rad = 180°/π ≈ 57.2958° ≈ 206,264.8″.
Quick Reference Table
| Turns (rev) | Radians (rad) |
|---|---|
| 1 rev | 6.28319 rad |
| 2 rev | 12.5664 rad |
| 5 rev | 31.4159 rad |
| 10 rev | 62.8319 rad |
| 25 rev | 157.08 rad |
| 50 rev | 314.159 rad |
| 100 rev | 628.319 rad |