This arc length calculator finds the length of a circular arc, the area of its sector, and the chord between its endpoints — from radius and central angle, in degrees or radians.
The radian is defined by this page: one radian is the angle whose arc equals the radius — which is why s = θr only works in radians. In degrees the same truth wears the conversion (θ/360 of the circumference). Chord = 2r sin(θ/2), always shorter than its arc — the straight-line tax on curved journeys.
s = θr — the Formula That Defines the Radian
Arc length is the circumference’s proportional slice: θ/2π of 2πr. The full circle’s own numbers live on our Circle Calculator. Part of our Math Calculators collection.
Worked Example (Verified)
- r = 10, θ = 60° (defaults): arc = (π/3) × 10 = 10.47; sector = ½ × (π/3) × 100 = 52.36; chord = 2 × 10 × sin 30° = 10.00 exactly — the 60° chord equals the radius (why hexagons tile so happily)
- Semicircle check: θ = 180°, r = 10 → arc 31.42 = πr ✔, sector 157.08 = half the circle’s area ✔ — formulas that reduce correctly at the extremes
- Clock arithmetic: a minute hand (r = 15 cm) sweeps 6° per minute → tip travels 1.57 cm/min, 94.2 cm per hour around the dial
- Earth flavor: 1° of latitude arc ≈ 111 km (r = 6,371 km × π/180) — the definition the nautical mile (1 arc-minute ≈ 1.852 km) was born from
Arc vs Sector vs Segment
- Arc: the curve itself (1-dimensional) — this page’s headline.
- Sector: the pie slice (arc + two radii) — area ½θr².
- Segment: between chord and arc — sector minus the triangle (½r²(θ − sin θ)); the sliver carpenters actually cut.
- Sagitta (segment height) = r − r cos(θ/2) — the number arch builders reach for.
Related Tools
- Circle Calculator — radius, circumference, area
- Trigonometry Calculator — the angle functions behind the chord
- Math Calculators — the full collection
Frequently Asked Questions
How do you find arc length?
s = θr with θ in radians. In degrees: s = (θ/360) × 2πr. For r = 10 and θ = 60°: (60/360) × 62.83 = 10.47. This calculator accepts either unit and also returns the sector area and chord.
What is the formula for sector area?
A = ½θr² (radians) — the same proportional logic: θ/2π of πr². At 60° and r = 10: ½ × 1.047 × 100 = 52.36 square units, exactly one-sixth of the full circle, as a 60° slice must be.
Why must the angle be in radians for s = θr?
Because the radian is defined so this is true: 1 radian subtends an arc of exactly one radius. Degrees are an arbitrary 360-part division (Babylonian legacy), so they need the 2π/360 conversion factor — the formula isn’t simpler in radians by luck, it’s true by definition.
What’s the difference between arc length and chord length?
The arc follows the curve; the chord cuts straight across (2r sin(θ/2)). The chord is always shorter — at 60° with r = 10: chord 10.00 vs arc 10.47. The gap grows with angle: at 180° the chord is the diameter (20) while the arc is 31.42.
How long is 1 degree of arc on Earth?
About 111 km at the equator for latitude (Earth radius ≈ 6,371 km × π/180 ≈ 111.2 km). Longitude degrees shrink toward the poles by cos(latitude). This is why the nautical mile is defined as one arc-minute of latitude: 1.852 km.
Is this arc length calculator free?
Yes — free, instant, no sign-up.