📏 Arc Length Calculator
Calculate arc length instantly with our free, comprehensive arc length calculator. Find the length of a circular arc from radius and central angle in degrees or radians. Perfect for geometry students, engineers, architects, manufacturers, and anyone working with circular segments, curved paths, or arc measurements in construction, design, or mathematical applications.
📋 How to Use
- Enter radius: Type the radius of the circle containing the arc.
- Select angle unit: Choose degrees or radians for your angle measurement.
- Enter angle: Type the central angle that defines the arc.
- Calculate: Click “Calculate Arc Length” for instant results.
- View results: See arc length, sector area, chord length, and angle conversions.
🔍 Understanding Arc Length
Arc Length (degrees): L = (θ/360) × 2πr
Arc Length (radians): L = θr
Where θ = central angle, r = radius
Arc length is the distance along a curved section of a circle’s circumference. An arc is defined by its radius and central angle. For a full circle (360° or 2π radians), arc length equals full circumference (2πr). For partial circles, arc length is proportional to the angle. A 90° arc (quarter circle) has length (90/360) × 2πr = 0.25 × circumference. Understanding arc length is essential for curved paths, circular segments, and any application involving partial circular measurements.
Degrees vs Radians in Arc Calculations
Angles can be measured in degrees (circle = 360°) or radians (circle = 2π ≈ 6.28). The radian formula (L = θr) is simpler – just multiply angle by radius. The degree formula requires converting to a fraction of 360° first. To convert degrees to radians: radians = degrees × π/180. To convert radians to degrees: degrees = radians × 180/π. 90° = π/2 rad, 180° = π rad. Radians are standard in calculus and advanced mathematics; degrees are more intuitive for everyday use.
Sector Area Related to Arc Length
A sector is the “pie slice” region bounded by an arc and two radii. Sector area relates to arc length: Area = (L × r)/2 or Area = (θ/360) × πr² (degrees) or Area = (θr²)/2 (radians). For radius 10 and 60° angle: arc length ≈ 10.47, sector area ≈ 52.36. The sector area and arc length increase proportionally with angle – doubling angle doubles both arc length and sector area.
Chord Length vs Arc Length
Chord is the straight-line distance between arc endpoints. Arc length is the curved distance along the circle. For same angle and radius, arc length is always longer than chord length (shortest path is straight). For 90° arc with radius 10: arc length ≈ 15.71, chord ≈ 14.14. As angle approaches 180°, chord approaches diameter (2r) while arc approaches half circumference (πr). For small angles, arc and chord are nearly equal; for large angles, they differ significantly.
Applications in Engineering and Design
Engineers calculate arc lengths for curved roads, railway tracks, conveyor belts, and mechanical linkages. Architects design arched doorways, windows, and architectural features requiring arc calculations. Manufacturers produce curved components with specified arc dimensions. Road designers calculate curve lengths for highway design and signage placement. CNC machining programs use arc lengths for tool path programming in manufacturing curved parts.
Navigation and Geography
Arc length calculations determine distances along curved Earth surfaces. Great circle routes (shortest paths between points on spheres) use arc length formulas with Earth’s radius and angular separation. Latitude/longitude calculations convert angular coordinates to actual distances using arc length principles. Sailors and pilots calculate course distances using spherical arc length calculations with Earth’s radius (approximately 6,371 km).
📊 Arc Length Examples
| Radius | Angle (degrees) | Angle (radians) | Arc Length |
|---|---|---|---|
| 10 | 90° | π/2 | 15.71 |
| 5 | 60° | π/3 | 5.24 |
| 15 | 120° | 2π/3 | 31.42 |
| 8 | 45° | π/4 | 6.28 |
| 12 | 180° | π | 37.70 |
| 20 | 30° | π/6 | 10.47 |
✨ Benefits
⚡ Instant Results
Calculate arc length immediately from radius and angle.
🎯 Dual Angle Units
Works with both degrees and radians for flexibility.
📊 Complete Info
See arc length, sector area, and chord length together.
🔧 High Precision
Accurate calculations using precise π and trigonometric values.
📱 Mobile Optimized
Calculate on any device for field work or homework.
🆓 Completely Free
Unlimited calculations with no registration required.
❓ FAQ
What is the formula for arc length?
Arc Length = (θ/360) × 2πr for degrees, or L = θr for radians, where θ is central angle and r is radius. For radius 10 and 60°: L = (60/360) × 2π(10) = (1/6) × 20π ≈ 10.47 units.
How do you measure an arc?
Place flexible tape measure along the curved arc path from start to end point. For calculation, measure radius and central angle, then use formulas. Arc length equals the fraction of full circumference corresponding to the angle fraction.
What’s the difference between arc length and sector area?
Arc length is linear distance along curve (meters, feet). Sector area is two-dimensional space enclosed by arc and two radii (square meters, square feet). Both increase with angle, but one measures length, the other area.
How do you convert degrees to radians?
Multiply by π/180. For 90°: 90 × π/180 = π/2 ≈ 1.571 radians. Radians to degrees: multiply by 180/π. Radians often simplify formulas – arc length in radians is just θr without the 360° conversion factor.
Can arc length be longer than circumference?
No, arc length is part of the circumference. Maximum arc length equals full circumference (at 360° or 2π radians). For angles exceeding 360°, you’re measuring multiple complete circles plus a partial arc.
How is chord length different from arc length?
Chord is straight-line distance between arc endpoints. Arc is curved distance along circle. Arc length is always ≥ chord length. For 180° (semicircle), chord = diameter, arc = half circumference (πr). For 90°, significant difference; for tiny angles, nearly equal.
What if I only know arc length and need radius or angle?
Rearrange formulas: r = L/(θ in radians) or θ(degrees) = (L/r) × (180/π). If arc length is 15.71 and angle is 90°, radius = 15.71/(π/2) ≈ 10. One formula needs two of three variables (r, θ, L) to find the third.