Arc Length Calculator - Find Curve Length ∫ds Fast

Arc length is the distance measured along the curved portion of a circle between two points. Unlike a chord (the straight line connecting the same two points), arc length follows the actual curve of the circumference. The arc length formula depends on whether you measure the central angle in degrees or radians:

• •Radians: Arc Length = radius × angle → s = rθ
• •Degrees: Arc Length = (θ/360) × 2πr
• •Percentage: Arc Length = (p/100) × circumference

The radian formula s = rθ is the most fundamental because a radian is defined as the angle subtended when the arc length equals the radius. This means 1 radian corresponds to an arc exactly as long as the radius.

Follow these steps to calculate arc length by hand, or let our calculator handle them for you:

1. Identify the radius (r) — the distance from the center to any point on the circle. If you have the diameter, divide by 2; if you have the circumference, divide by 2π.
2. Identify the central angle (θ) — the angle formed at the center of the circle between two radii that define the arc.
3. Convert to radians if in degrees: θ_rad = θ° × (π/180).
4. Apply the formula: s = r × θ_rad.

Example 1: Arc Length from Degrees

A circle has radius 10 cm and a central angle of 60°.
θ_rad = 60 × π/180 = π/3 ≈ 1.0472 rad
s = 10 × π/3 = 10π/3 ≈ 10.472 cm

Example 2: Arc Length in Terms of π

Quarter-circle (90°) with radius 8 m:
θ_rad = 90 × π/180 = π/2
s = 8 × π/2 = 4π ≈ 12.566 m

Many textbooks and exams expect answers in terms of π. This calculator shows the π form automatically when the result is a clean multiple of π.

Example 3: Finding the Central Angle from Arc Length and Radius

Arc length s = 15 units, radius r = 6 units.
θ = s / r = 15 / 6 = 2.5 rad ≈ 143.24°

A chord is the straight-line segment joining the two endpoints of an arc, while the arc length follows the curve. The chord is always shorter than (or equal to) the arc. The chord length formula is:

c = 2r × sin(θ/2)

For a 60° arc with radius 10, the chord is 2 × 10 × sin(30°) = 10, while the arc length is 10π/3 ≈ 10.47 — close but slightly longer because the arc curves outward. For details on chord geometry, see our chord calculator. To convert between arc length and chord length, enter any two of the three values (radius, angle, arc length) and read the chord from the results.

Every arc is part of the circle's total circumference (C = 2πr). The fraction of the circumference that an arc represents equals the fraction of the full angle it subtends. This connection extends to the sector area:

• •Sector area = ½r²θ (radians) or (θ/360) × πr²
• •Arc length = rθ (the "perimeter" of the curved edge of the sector)

Our calculator outputs both automatically. If you need the full circumference, use our circumference calculator, or note that the circumference is simply the arc length when θ = 360° = 2π.

For curves that aren't circular, the arc length is computed with integration. Given a function y = f(x) on the interval [a, b]:

L = ∫_a^b √[1 + (dy/dx)²] dx

For parametric curves x = f(t), y = g(t):

L = ∫_a^b √[(dx/dt)² + (dy/dt)²] dt

The circular arc formula s = rθ is a special case of the parametric integral: parametrize the circle as x = r cos t, y = r sin t, and integrate from 0 to θ. The result simplifies to rθ because x′² + y′² = r².

For evaluating these calculus-based arc length integrals numerically, try our definite integral calculator.

Understanding the relationship between angle units and arc length is essential:

Use this table as a quick reference, or enter any angle into our calculator and it will convert between all three units automatically. For more on angle conversions and measurement, see our angle calculator.

Arc length calculations appear everywhere in engineering, construction, and everyday life:

• Construction & Landscaping: Curved retaining walls, driveways, and walkways. A 90° curved path with a 20-foot radius needs exactly 10π ≈ 31.42 feet of material.
• Engineering & Manufacturing: Gear tooth profiles, cam mechanisms, conveyor belt paths, and robotic arm trajectories all require precise arc measurements.
• Navigation & Geography: Great-circle distances between cities on Earth are arc lengths on a sphere. Airlines use these calculations for flight path optimization.
• Education: From geometry classes to calculus courses, arc length is a core topic. Students use it to connect trigonometry, integration, and coordinate geometry. See our geometry calculator for related tools.
• Sports Tracks: Running tracks use staggered starts calculated from arc length differences between lanes with slightly different radii.

This free online arc length calculator lets you find the arc length, central angle, or radius of any circular arc. It supports degrees, radians, and percentages, shows results in terms of π, computes the chord length and sector area, and provides a visual arc diagram with step-by-step solutions. Whether you need it for homework, construction, or engineering, bookmark this page for instant, reliable arc calculations.