units
Enter a positive radius.Any unit — result in same unitdeg
Enter angle between 0.001 and 360.Arc Length
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Sources & Methodology
Arc length formula L = rθ from standard Euclidean geometry. Sector area = ½r²θ. Chord = 2r⋅sin(θ/2). Verified against Khan Academy and NCTM standards.
Khan Academy — Arc Length from Central Angle
Confirms arc length formula in degrees L = (θ/360) × 2πr and radians L = rθ used to verify this calculator.
NCTM — Principles and Standards for School Mathematics
Curriculum reference for circle geometry standards including arc length, sector area, and radian measure.
Arc (degrees): L = (θ°/360) × 2πr • Arc (radians): L = r × θ
Sector Area: A = ½ × r² × θ • Chord: c = 2r × sin(θ/2) • Circumference: C = 2πr
Sector Area: A = ½ × r² × θ • Chord: c = 2r × sin(θ/2) • Circumference: C = 2πr
⏱ Last reviewed: April 2026
How Is Arc Length Calculated?
An arc is a curved portion of a circle’s circumference. Arc length depends on two values: the radius and the central angle. In radians, arc length equals radius times angle directly. In degrees it becomes a fraction of the total circumference.
The Arc Length Formula
Radians: L = r × θ
Degrees: L = (θ / 360) × 2πr
Sector: A = ½ × r² × θ Chord: c = 2r × sin(θ/2)
Example: r=10, θ=60° → Radians: 60 × π/180 = 1.0472 → Arc = 10 × 1.0472 = 10.472 units
Common Angles Reference Table
| Angle | Radians | Arc (r=1) | Arc (r=10) | Sector (r=10) |
|---|---|---|---|---|
| 30° | π/6 | 0.524 | 5.236 | 26.18 |
| 45° | π/4 | 0.785 | 7.854 | 39.27 |
| 60° | π/3 | 1.047 | 10.472 | 52.36 |
| 90° | π/2 | 1.571 | 15.708 | 78.54 |
| 180° | π | 3.142 | 31.416 | 157.08 |
| 270° | 3π/2 | 4.712 | 47.124 | 235.62 |
| 360° | 2π | 6.283 | 62.832 | 314.16 |
Why Radians Simplify the Formula
A radian is defined as the angle producing an arc equal in length to the radius. So L = r × θ is a direct product with no conversion factor needed. The degree formula requires π/180 because degrees are an arbitrary subdivision of the circle.
Arc Length vs Chord Length
Arc length is the curved path; chord is the straight line between the same endpoints. Arc is always longer. At 180°: arc = πr ≈ 3.14r but chord = 2r. For angles under 5° they are nearly identical.
💡 Degree to Radian Quick Reference: 30°=0.524 • 45°=0.785 • 90°=1.571 • 180°=3.14159 • 360°=6.283. Multiply degrees by π/180 to convert.
Frequently Asked Questions
Arc Length = r × θ (with θ in radians). In degrees: L = (θ/360) × 2πr. Example: r=5, θ=90°. Convert: 90 × π/180 = π/2. Arc = 5 × 1.5708 = 7.854 units.
L = (θ°/360) × 2πr. Divide angle by 360 to get fraction of circle, multiply by circumference. Example: 60°, r=10: (60/360) × 2π × 10 = 10.472 units.
Multiply: L = r × θ. Example: r=8, θ=π/4 radians: Arc = 8 × 0.7854 = 6.283 units. One radian is defined as the angle producing an arc equal in length to the radius.
A radian is the angle at the center of a circle when arc length equals the radius. One radian ≈ 57.296°. A full circle = 2π radians = 360°. Radians simplify arc length, trig derivatives, and rotational formulas.
Sector Area = ½ × r² × θ (θ in radians). Or: (θ°/360) × πr². Example: r=6, θ=60°: Area = (60/360) × π × 36 = 18.85 sq units.
Chord = 2r × sin(θ/2) where θ is in radians. Example: r=10, θ=90°: Chord = 2 × 10 × sin(45°) = 20 × 0.7071 = 14.142 units.
Arc length = curved distance. Chord = straight-line distance. Arc is always longer. At 180°: arc = πr ≈ 3.14r, chord = 2r. For angles under 10° the two are nearly equal.
Multiply by π/180 ≈ 0.01745. Key values: 30°=0.524, 45°=0.785, 60°=1.047, 90°=1.571, 180°=3.14159, 360°=6.283. To convert back: multiply radians by 180/π ≈ 57.296.
Arc = (θ/360) × Circumference. A 90° arc = ¼ circumference. A 180° arc = ½ circumference. A 360° arc = full circumference = 2πr.
Arc length applies in engineering (belt length around pulleys, road curve design), architecture, CNC machining, animation, astronomy, and physics. L = rθ is fundamental in any rotational system and circular motion.
In calculus: L = ∫ √(1 + (dy/dx)²) dx from a to b. For a circle this evaluates exactly to rθ. The calculus version extends arc length to any smooth curve.
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