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Revolution — Definition, Formula & Examples

A revolution is one complete turn around a circle, bringing you back to the starting point. It equals 360 degrees or 2π radians.

A revolution (rev) is a unit of angular measure defined as one full rotation about a fixed axis, equivalent to an angular displacement of exactly 2π2\pi radians or 360°360°.

Key Formula

θrev=θdeg360=θrad2π\theta_{\text{rev}} = \frac{\theta_{\text{deg}}}{360} = \frac{\theta_{\text{rad}}}{2\pi}
Where:
  • θrev\theta_{\text{rev}} = Angle measured in revolutions
  • θdeg\theta_{\text{deg}} = Angle measured in degrees
  • θrad\theta_{\text{rad}} = Angle measured in radians

How It Works

To convert from revolutions to degrees, multiply by 360. To convert from revolutions to radians, multiply by 2π2\pi. Going the other direction, divide degrees by 360 or divide radians by 2π2\pi to get revolutions. For example, a half revolution is 0.5×360°=180°0.5 \times 360° = 180° or 0.5×2π=π0.5 \times 2\pi = \pi radians.

Worked Example

Problem: Convert 3.5 revolutions to degrees and to radians.
Convert to degrees: Multiply the number of revolutions by 360.
3.5×360°=1,260°3.5 \times 360° = 1{,}260°
Convert to radians: Multiply the number of revolutions by 2π.
3.5×2π=7π≈21.99 rad3.5 \times 2\pi = 7\pi \approx 21.99 \text{ rad}
Answer: 3.5 revolutions = 1,260° = 7π radians.

Visualization

Why It Matters

Revolutions appear constantly in physics and engineering — rpm (revolutions per minute) describes motor speeds, wheel rotations, and turbine rates. Converting fluently between revolutions, degrees, and radians is essential in trigonometry and any course involving circular motion.

Common Mistakes

Mistake: Confusing 1 revolution with π radians instead of 2π radians.
Correction: One full revolution covers the entire circle, which is 2π radians (≈ 6.28 rad). The value π radians corresponds to only half a revolution (180°).

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