Arc Length (Radians)
Arc length from radius and central angle in radians.
Formula
Variables
Example
r=2, \theta=π: s=2π
How it works
For a circle, an angle measured in radians tells you how much of the circle you have turned through compared with the radius. One full revolution is 2π radians and corresponds to a distance of 2πr around the circle, so each radian accounts for exactly r units of arc length. That is why multiplying the radius by the central angle in radians gives the length of the curved section.
When to use it
Use this when you know a circle’s radius and the central angle, and you need the length of the arc between two points. It is common in geometry, trigonometry, physics, and any problem involving circular motion or sector lengths.
Worked example, step by step
Given: r = 5 cm, θ = 1.2 rad Use the formula: s = rθ Substitute the values: s = 5 × 1.2 Calculate: s = 6 Answer: the arc length is 6 cm
Common mistakes to avoid
- Using degrees instead of radians in the formula
- Multiplying by diameter instead of radius
- Forgetting to include units in the final answer
Did You Know?
Measuring angles in radians makes arc length simply radius times angle — one reason radians dominate higher mathematics.
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