Arc length is the distance measured along the curved portion of a circle. This calculator determines the arc length using the radius and central angle, accepting both degrees and radians.
Formula
The arc length is calculated as:
[\text{Arc Length} = r \times \theta]
Where:
- r is the radius of the circle
- ฮธ (theta) is the central angle in radians
Converting Degrees to Radians
If your angle is in degrees, convert it first:
[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}]
Calculation Example
For a circular pathway with radius 5 meters and central angle of 60 degrees:
Step 1: Convert to radians
[\text{Radians} = 60 \times \frac{\pi}{180} = 1.047 \text{ radians}]
Step 2: Calculate arc length
[\text{Arc Length} = 5 \times 1.047 = 5.235 \text{ meters}]
Common Arc Lengths
| Central Angle | Fraction of Circumference | Arc Length Formula |
|---|---|---|
| 360ยฐ (2ฯ rad) | Full circle | 2ฯr |
| 180ยฐ (ฯ rad) | Half circle | ฯr |
| 90ยฐ (ฯ/2 rad) | Quarter circle | ฯr/2 |
| 60ยฐ (ฯ/3 rad) | Sixth of circle | ฯr/3 |
Applications
- Architecture: Designing curved structures and arches
- Engineering: Calculating wheel rotations and gear teeth
- Road Design: Planning curved highway sections
- Astronomy: Measuring angular distances in the sky
- Sports: Analyzing curved ball trajectories