Arc Length Calculator

**Arc length (radians):** L = r × θ Where: - L = arc length - r = radius - θ = central angle in radians **Arc length (degrees):** L = (θ / 360) × 2πr = (πrθ) / 180 **Conversions:** - 1 radian ≈ 57.296° - 1° ≈ 0.01745 radians - Full circle: 2π rad = 360° **Worked example: quarter circle** Radius r = 10, angle = 90°. Convert: 90° = π/2 ≈ 1.571 rad. L = 10 × 1.571 = 15.71 Or in degrees: L = (90/360) × 2π × 10 = (1/4) × 62.832 ≈ 15.708. Either method gives same result. **Full circumference check:** For full circle: θ = 2π rad = 360°. L = r × 2π = 2πr (the familiar circumference formula). **Common angles and arc lengths (r = 1):** | Angle | Degrees | Radians | Arc length (r=1) | |---|---|---|---| | 30° | 30 | π/6 ≈ 0.524 | 0.524 | | 45° | 45 | π/4 ≈ 0.785 | 0.785 | | 60° | 60 | π/3 ≈ 1.047 | 1.047 | | 90° | 90 | π/2 ≈ 1.571 | 1.571 | | 120° | 120 | 2π/3 ≈ 2.094 | 2.094 | | 180° | 180 | π ≈ 3.142 | 3.142 | | 270° | 270 | 3π/2 ≈ 4.712 | 4.712 | | 360° | 360 | 2π ≈ 6.283 | 6.283 | For other radii, multiply by r. **Arc length for variable curves (calculus):** For a parametric curve x(t), y(t) from t=a to t=b: L = ∫_a^b √((dx/dt)² + (dy/dt)²) dt For y = f(x) from x=a to x=b: L = ∫_a^b √(1 + (dy/dx)²) dx For polar curves r = f(θ): L = ∫ √(r² + (dr/dθ)²) dθ **Chord vs arc:** For central angle θ (radians): - **Chord** (straight line): c = 2r × sin(θ/2) - **Arc length** (curved): L = r × θ For small angles, chord ≈ arc. For 180° (half circle): chord = 2r (diameter), arc = πr. **Sagitta (height of arc):** s = r × (1 − cos(θ/2)) Used in optics and engineering to describe arc dimensions. **Earth's great-circle distance:** Distance between two points on Earth at angular separation θ (rad): L = R × θ Where R ≈ 6,371 km (Earth's radius). For 1° of latitude: 6,371 × π/180 ≈ 111.2 km per degree. For 1' (minute): ≈ 1.853 km (close to 1 nautical mile by design). For 1" (second): ≈ 30.9 m. **Pulley/belt length:** For two pulleys radii r₁, r₂ connected by belt: - Arc on pulley 1: L₁ = r₁ × θ₁ - Arc on pulley 2: L₂ = r₂ × θ₂ - Straight runs between Total belt length depends on pulley positions and configurations. **Bicycle wheel distance:** Each wheel rotation = 1 circumference of arc = 2πr. For wheel radius 35 cm: circumference ~2.2 m. 100 RPM gives ~220 m/min = 13.2 km/h. **Speed on circular track:** Speed = (angular velocity) × (radius) = ω × r. For race car going 100 km/h on 200 m radius curve: - Convert: 100 km/h = 27.78 m/s. - Angular velocity: ω = v/r = 27.78/200 = 0.139 rad/s. - Going around full loop (2π rad): T = 2π/0.139 ≈ 45.2 s per lap. **Radian-degree conversions (key values):** | Radians | Degrees | |---|---| | π/6 | 30° | | π/4 | 45° | | π/3 | 60° | | π/2 | 90° | | π | 180° | | 3π/2 | 270° | | 2π | 360° | **Programming:** Most programming languages use radians for trig functions: - Math.sin(angle_in_radians) - For degrees: convert first or use degree-versions like sind() in MATLAB. **Common applications:** - **Engineering**: belt drives, pipe bends, curved structural elements. - **Navigation**: great-circle distances on Earth. - **Machining**: CNC tool paths for arcs. - **Architecture**: curved walls, domes, arches. - **Sports**: track design (banking on curved sections). - **Surveying**: curved property boundaries. - **Photography**: panoramic stitching (angular field of view). - **Mechanical engineering**: cam profiles, gear arc design. **Special case: full circumference:** L = 2πr (when θ = 2π radians = 360°) **Special case: semi-circle:** L = πr (when θ = π radians = 180°)