Arc Length Calculator

Understanding the Arc Length Formula

Arc length measures the distance along the curved edge of a circle between two endpoints defined by a central angle. Engineers, architects, and mathematicians apply this calculation daily — from laying out highway curves and programming CNC toolpaths to computing satellite orbital arcs and sizing decorative architectural elements.

The Core Formula

The arc length L of a circular arc follows the formula:

L = r · θ

where r is the radius of the circle and θ is the central angle expressed in radians. This relationship emerges directly from the definition of a radian: one radian is the angle subtended at the center when the arc length equals the radius. For any angle θ in radians, the arc scales proportionally, yielding an exact — not approximate — result for any circular arc.

Variable Breakdown

• Radius (r): The fixed distance from the circle's center to its edge. The computed arc length carries the same unit as the radius — millimeters, centimeters, meters, inches, or feet — with no additional conversion needed.
• Central Angle (θ): The angle at the circle's center subtended by the arc. Direct substitution into L = rθ requires radians. A full circle spans exactly 2π ≈ 6.2832 radians.
• Angle Unit: This calculator accepts degrees, radians, gradians, and turns, converting each to radians internally. One full revolution equals 360°, 2π radians, 400 gradians, or 1 turn.

Angle Unit Conversions

The calculator applies the following exact conversion factors before computing arc length:

• Degrees to radians: θ_rad = θ_deg × π/180. Example: 90° × π/180 = π/2 ≈ 1.5708 rad.
• Gradians to radians: θ_rad = θ_grad × π/200. Example: 100 grad × π/200 = π/2 rad.
• Turns to radians: θ_rad = θ_turns × 2π. Example: 0.25 turns × 2π = π/2 rad.

Formula Derivation

A full circle of radius r has circumference 2πr. A central angle of θ radians represents the fraction θ/(2π) of the complete circle. Multiplying that fraction by the full circumference yields:

L = (θ / 2π) × 2πr = rθ

This algebraic cancellation confirms the formula is exact for all circular arcs. For non-circular curves, calculus-based integral formulas extend this approach — as detailed in Paul's Online Math Notes: Arc Length (Calculus II) and Whitman College Calculus Online, Section 13.3: Arc Length and Curvature.

Worked Examples

Example 1 — Bicycle wheel: Radius = 35 cm; central angle = 120° = 2π/3 rad. Arc length: L = 35 × (2π/3) ≈ 35 × 2.0944 ≈ 73.3 cm.

Example 2 — Highway horizontal curve: Design radius = 500 m; central angle = 15°. Conversion: 15° × π/180 ≈ 0.2618 rad. Arc length: L = 500 × 0.2618 = 130.9 m. The Federal Highway Administration Field Formulas (M 22-24) specifies this exact method for horizontal curve layout on public roads, confirming its use at the highest engineering standard.

Example 3 — Clock minute hand: Hand length = 12 cm; 20-minute sweep = 120° = 2π/3 rad. Arc length: L = 12 × (2π/3) ≈ 25.1 cm.

Practical Applications

• Highway and civil engineering: Horizontal curve layout and road staking using arc lengths derived from design radii and deflection angles per FHWA standards. Engineers compute arc lengths to establish tangent-to-curve transitions and control stations along the centerline.
• Mechanical drives: Belt contact arc on pulleys determines friction surface area and torque transmission capacity. Larger contact arcs enable greater power transfer without slippage.
• Astronomy and navigation: Great-circle arc distances between geographic waypoints and arc lengths of orbital segments. Satellite ground tracks and flight path planning rely on precise arc length calculations.
• Architecture and construction: Sizing curved arches, circular staircases, barrel vaults, and decorative trim elements. Accurate arc lengths ensure proper fit and structural integrity of curved members.
• CNC and laser manufacturing: Toolpath programming for curved cuts and laser-etching arcs on circular stock. Machines require precise arc length specifications to achieve dimensional accuracy.
• Polar and parametric curves: Extended integral methods for non-circular arcs, described in depth by UT Austin Calculus — Arc Length of Polar Curves.