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Circle Arc Length Calculator
Calculate arc length, chord length, and related measurements
📚 Examples, Rules & Help
⚡Quick Examples of Arc Length
🔍How to Calculate Arc Length
Convert Angle to Radians
If angle is in degrees: θ(radians) = θ(degrees) × π/180. Radians make the arc length formula simpler.
Calculate Arc Length
Arc Length = θ × r, where θ is in radians. This gives the curved distance along the circle's edge.
Calculate Chord Length
Chord Length = 2r × sin(θ/2). This is the straight-line distance between the arc's endpoints.
Compare Arc vs Chord
Arc length is always greater than chord length for the same angle. The difference decreases as the angle gets smaller.
🌍Real-World Applications
🔧 Engineering
Curved pipe design, gear teeth
🏗️ Construction
Arch spans, curved roads
🏭 Manufacturing
Belt lengths, curved cuts
🧭 Navigation
Distance along curved paths
❓Frequently Asked Questions
What is arc length?
Arc length is the distance along the curved edge of a circle between two points. It's a portion of the circle's circumference, measured along the curve rather than as a straight line.
How do you calculate arc length?
Arc length = θ × r, where θ is the central angle in radians and r is the radius. For degrees: Arc length = (θ/360°) × 2πr.
What's the difference between arc length and chord length?
Arc length is the curved distance along the circle's edge, while chord length is the straight-line distance between the same two points. Arc length is always longer than chord length for the same angle.
When should I use degrees vs radians for arc length?
Radians are preferred for mathematical calculations as they make the formula simpler (s = θr). Use degrees for everyday problems, but convert to radians for precise calculations.
What's the relationship between arc length and sector area?
Both depend on the central angle and radius. Sector area = (1/2) × arc length × radius. They're proportional - double the angle, double both the arc length and sector area.
Can I find the central angle if I know arc length and radius?
Yes! Central angle (radians) = arc length ÷ radius. Convert to degrees by multiplying by 180/π.
🎯Common Use Cases
Construction & Infrastructure
- Road and highway curve design and length calculations
- Railway track arc measurements for smooth turns
- Bridge arch span calculations and material estimates
- Tunnel boring calculations for curved underground passages
Manufacturing & Industrial
- Belt and chain length calculations for machinery
- Pipe bending measurements for plumbing systems
- Wire and cable length estimates for curved routing
- Sheet metal forming calculations for automotive parts
Navigation & Sports
- GPS route calculations along curved paths
- Athletic track lane measurements and distances
- Golf course design and hole distance planning
- Hiking trail distance calculations for curved routes
Arc vs Chord Relationships
- Arc length > chord length for the same angle (always)
- Difference decreases as central angle decreases
- Small angles: arc ≈ chord (useful approximation < 10°)
- Half circle: arc = πr, chord = 2r (ratio ≈ 1.57)
💡Calculator Tips & Best Practices
💡Arc vs Chord Selection
Arc length measures the actual curved distance traveled, while chord length is the straight-line shortcut. Choose based on your application.
📏Material Estimation Accuracy
Always use arc length (not chord) for material calculations like cables, pipes, or roads. The difference can be significant for large angles.
⭐Small Angle Approximation
For angles less than 10°, arc length ≈ chord length. This can simplify quick calculations when high precision isn't needed.
⚠️Radian Calculations
Arc length formula s = θr only works with radians. Always convert degrees to radians first: rad = deg × π/180.
📝Circumference Percentage
Arc length as percentage of full circumference helps visualize the proportion: percentage = (arc length / 2πr) × 100%.
⭐Construction Precision
For construction projects, calculate arc lengths to 3-4 decimal places to account for material waste and ensure proper fit.
📚 References & Further Reading
Physics applications of arc length in circular motion
External Link
Standards for precise angular and length measurements
External Link
Note: These references provide additional mathematical context and verification of the formulas used in this calculator.