Arc Length
Arc length measures the distance along a curved path of a circle. The formula L = r × θ (with θ in radians) connects radius and central angle to the curved distance—essential for railroad curves, sports tracks, and compass bearings.
Why This Mathematical Concept Matters
Why: Arc length appears in railroad curves, sports track design, compass bearings, and any curved path. Understanding L = rθ reveals why 1 radian subtends an arc equal to the radius.
How: Use L = r × θ when θ is in radians. For degrees: L = (θ/360°) × 2πr. Chord length c = 2r sin(θ/2). Sector area A = (θ/360°) × πr².
- ●1 radian subtends an arc equal to the radius—the natural angle unit for circles.
- ●Olympic track lanes use semicircles of radius 36.5 m for the 400 m track.
- ●As θ → 0, arc length and chord length converge—nearly equal for small angles.
Arc Length — From Radius and Central Angle
Enter radius and angle (degrees or radians) to get arc length, sector area, chord length, and arc-to-chord ratio.
🔵 Sample Examples — Click to Load
Input
Arc Properties (Bar)
Arc vs Remaining Circumference
Properties Radar (Scaled)
Step-by-Step Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
L = r × θ (radians)—arc length equals radius times angle.
— Standard formula
Chord length c = 2r sin(θ/2). Arc length ≥ chord always.
— Geometry
Key Takeaways
- Arc length formula: L = r × θ — where θ is in radians. For degrees: L = (θ/360°) × 2πr
- Chord length: c = 2r sin(θ/2) — the straight-line distance between arc endpoints
- Arc length is always greater than or equal to chord length — the chord is the shortest path
- Sector area: A = (θ/360°) × πr² — proportional to the central angle
- Full circumference: C = 2πr — arc length when θ = 360° (2π radians)
Did You Know?
How Arc Length Works
Arc length is the distance along the curved part of a circle between two points defined by a central angle.
Formula: L = r × θ (radians)
When the angle is in radians, arc length equals radius times angle. This is because 1 radian subtends an arc equal to the radius.
Formula: L = (θ/360°) × 2πr (degrees)
For degrees, convert: θ_rad = θ_deg × π/180. Then L = r × θ_rad. Equivalently, L = (θ/360°) × circumference.
Chord vs Arc
The chord is the straight line between endpoints. Arc length ≥ chord length always. For small angles they are nearly equal.
Sector Area
The sector (pie slice) area is A = (θ/360°) × πr² = ½ r² θ (radians). It is proportional to the central angle.
Expert Tips
Use Radians for Calculus
In calculus and physics, radians are preferred. dL/dθ = r when θ is in radians — no conversion factors.
Check Your Angle Unit
Mixing degrees and radians is a common error. 90° = π/2 rad ≈ 1.57 rad — very different values!
Arc-to-Chord Ratio
The ratio L/c = θ/(2 sin(θ/2)) increases with angle. For θ → 0 it approaches 1.
Real-World Applications
Railroad curves, sports tracks, compass arcs, pendulum swings, and curved road design all use arc length.
Formula Comparison
| Quantity | Formula | When to Use |
|---|---|---|
| Arc Length | L = r × θ | θ in radians |
| Arc Length | L = (θ/360°) × 2πr | θ in degrees |
| Chord Length | c = 2r sin(θ/2) | Straight-line distance |
| Sector Area | A = ½ r² θ | θ in radians |
Frequently Asked Questions
What is the arc length formula?
L = r × θ when θ is in radians. For degrees: L = (θ/360°) × 2πr. The arc length is proportional to both radius and central angle.
What is the difference between arc length and chord length?
Arc length is the distance along the curved path. Chord length is the straight-line distance between the same two points. Arc length ≥ chord length always.
Why use radians instead of degrees?
Radians give a simpler formula (L = rθ) and are the natural unit in calculus. 1 radian = 180°/π ≈ 57.3°.
How do I find the central angle from arc length?
Rearrange: θ = L/r (radians) or θ = (L/r) × (180°/π) (degrees).
What is the arc length of a semicircle?
For a semicircle (180°), L = πr — half the circumference. The chord equals the diameter (2r).
Can arc length exceed the circumference?
For a single arc, the central angle is typically 0° to 360°. Arc length = (θ/360°) × 2πr, so it cannot exceed 2πr.
How is arc length used in engineering?
Railroad curves, road design, pipe bending, gear teeth, and curved structural elements all require arc length calculations.
How accurate is this calculator?
Uses JavaScript Math with ~15 significant digits. Results are rounded for display. Verify for critical applications.
Arc Length by the Numbers
Official & Trusted Sources
Disclaimer: This calculator provides mathematically precise results based on standard formulas L = rθ, c = 2r sin(θ/2), and sector area formulas. Results are limited by floating-point precision. For critical engineering or surveying applications, verify with domain-specific tools. Not a substitute for professional analysis.