Arc Length Calculator

Definition

The arc length of a circle is the length of the curve that is part of a circle of radius r > 0 corresponding to a central angle θ. When the angle is measured in radians, the formula is especially simple: L = r · θ. When the angle α is given in degrees, the same arc has length L = 2πr · (α/360) = πr · (α/180). The two formulas are equivalent because θ = α · (π/180).

Geometrically the arc is the image of an angular sector on the circle. A full circle corresponds to θ = 2π (or α = 360°) and yields L = 2πr — the classical circumference. MathWorld (Circular Arc) and Euclidean geometry texts treat arc length as proportional to the central angle at fixed r. This calculator computes L from given r and angle. The requirement r > 0 is geometric: a negative “radius” does not describe a circle in the Euclidean plane.

Historical and conceptual context

Already in Euclid’s Elements (Book III) central angles and arcs are studied; the ratio of arcs on the same circle equals the ratio of central angles. Archimedes, in the measurement of a circle, related circumference to diameter, leading to the constant π. The modern definition of a radian — the angle for which arc length equals the radius — makes L = rθ almost tautological with the angle measure.

Conceptually, arc length is not the area of a sector (that is (1/2)r²θ) nor the chord (the straight segment between the arc’s endpoints). The chord has length 2r sin(θ/2) and is always less than or equal to the arc (equality only in the degenerate case θ = 0). This calculator returns only the length of the circular curve.

Formal properties

Inputs: r > 0 (radius), θ ∈ ℝ in radians or α in degrees. Output: L ≥ 0 for a nonnegative angle; for a negative angle applications often take |θ| or |α|. Homogeneity: L scales linearly with r and with |θ|.

Equivalent formulas: L = rθ (θ in rad); L = 2πr · (α/360) = πrα/180 (α in degrees); L = (θ/(2π)) · 2πr — the fraction of the full circumference. For θ = π (semicircle) L = πr; for θ = π/2 (quarter) L = πr/2; for θ = 2π, L = 2πr.

Differentially: the arc element on the circle x = r cos φ, y = r sin φ satisfies ds = r dφ, so integrating from 0 to θ gives ∫₀^θ r dφ = rθ. This derivation is standard in calculus and matches the radian definition of angle measure.

Full derivation of the formula L = rθ

Step 1. The circumference of a full circle is C = 2πr (definition of π as circumference over diameter, or Archimedes in modern form).

Step 2. A full angle is 2π radians. With uniform arc density on the circle, the length for angle θ is proportional: L / C = θ / (2π), hence L = 2πr · θ/(2π) = rθ.

Step 3 (degrees). A full angle is 360°. L / C = α/360, so L = 2πr · (α/360). Since θ = απ/180, substitution yields L = rθ — unit consistency.

Step 4 (parametrization). Let γ(φ) = (r cos φ, r sin φ), φ ∈ [0, θ]. Then |γ′(φ)| = r, so the curve length ∫₀^θ |γ′| dφ = rθ. The same with |θ| for the opposite orientation.

Special cases

Example: r = 10, θ = π/3 (60°) → L = 10 · π/3 ≈ 10.472. The same from the degree formula: L = 2π·10·(60/360) = 10π/3. For r = 1, θ = 1 rad → L = 1 — the definition of a radian. For α = 360°, L = 2πr.

As r → 0 the arc vanishes. For very small |θ|, L ≈ r|θ| remains linear; the chord 2r sin(θ/2) ≈ r|θ| to first order, so for small angles arc and chord are close. When |θ| > 2π, L = rθ still gives a multi-wrap length — physical meaning depends on whether arcs longer than the circumference are allowed.

Numerical and computational notes

The calculator must know whether the angle input is in degrees or radians — unit confusion is the most common error (e.g. using 90 instead of π/2). Computation: if degrees, L = 2 * π * r * (α/360); if radians, L = r * θ. Require r > 0.

A comma or a period is accepted equivalently. The length unit of the result matches the unit of r (metres, cm, px — no conversion). For very large r·|θ|, floating-point arithmetic may lose relative precision, but for typical engineering data it is sufficient.

Related concepts and authorities

Related tools: circumference of a circle (θ = 2π), sector area ((1/2)r²θ), chord length, degrees–radians converter, and central angle. Arc length does not replace a line integral along an arbitrary curve — here the curve is a circle.

Authorities: Euclid, Elements, Book III (arcs and central angles); Archimedes, Measurement of a Circle; MathWorld — Circular Arc, Radian; NIST (π and angle conversions in the DLMF). The calculator does not replace a geometry textbook. Separately compute the circumference 2πr and compare the fraction α/360 with the ratio L/(2πr) — a quick check that angle units are consistent.