Arc Length Calculator | Curves & Circle Arcs
Calculate arc length for any function f(x) over an interval using numerical integration, or compute a circle arc length from radius and angle. Also shows sector area, chord length, and step-by-step working.
Quick examples
What Is the Arc Length Calculator | Curves & Circle Arcs?
Arc length measures the actual distance along a curve, not the straight-line distance between two points, but the length you would get if you laid the curve flat. This calculator handles two types of arc length problems: computing the length of a curve defined by a function f(x), and computing the arc of a circle from its radius and central angle.
- ▸Function mode: Enter any f(x) expression (x^2, sin(x), sqrt(x), log(x), exp(x)) and the integration interval [a, b]. The derivative is computed numerically and the arc length integral is evaluated via Simpson's rule with 1000 subintervals.
- ▸Circle arc mode: Enter radius and central angle (degrees or radians). Also shows sector area, chord length, and segment area.
- ▸Quick examples: One-click presets for common functions (y = x², sin(x), √x, ln(x)) to see results immediately.
- ▸Step-by-step output: Every result includes the formula used, the approach taken, and key intermediate values.
Formula
L = ∫ₐᵇ √(1 + [f′(x)]²) dx
Derived from Pythagoras: each tiny segment ds = √(dx² + dy²) = √(1 + (dy/dx)²) dx. Summing all segments gives the total arc length.
L = r × θ (θ in radians)
One radian is defined as the angle where arc length = radius. So for any angle θ, L = rθ. Simple and exact, no integration needed.
f′(x) ≈ [f(x+h) − f(x−h)] / 2h (central difference, h = 10⁻⁵)
Since f′(x) is needed inside the integral and is hard to compute symbolically for arbitrary functions, it is approximated numerically.
L ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]
n = 1000 subintervals. Simpsons rule has O(h⁴) error, far more accurate than the trapezoid rule for smooth functions.
How to Use
- 1
Choose Function or Circle Arc mode
Function mode integrates the arc length formula along a curve. Circle arc mode uses the exact formula L = rθ and also calculates sector area, chord, and segment.
- 2
Enter the function (Function mode)
Type f(x) using standard notation: x^2 for x², sin(x), cos(x), sqrt(x), log(x) for natural log, exp(x) for eˣ. Do not use * for multiplication, write 2*x or 2x.
- 3
Set the bounds a and b
Enter the x-values that define the start and end of the curve. a must be less than b. Use decimal approximations for irrational bounds (e.g. 3.14159 for π).
- 4
For circle arc: enter radius and angle
Enter the circle's radius and the central angle. Toggle between degrees and radians. Use the common angle presets (90°, 180°, etc.) for quick calculations.
- 5
Click Calculate and read the results
Results include the arc length, plus chord length, sector area (circle mode), or function values at the bounds (function mode). The step-by-step panel explains each part of the calculation.
Example Calculation
Example 1 | Arc length of y = x² from 0 to 2
Example 2 | Quarter circle arc (r = 5, θ = 90°)
Example 3 | Arc length of y = sin(x) from 0 to π
The arc of a half-sine wave is about 21% longer than π (the horizontal distance).
Understanding Arc Length | Curves & Circle Arcs
What Is Arc Length?
Arc length is the distance measured along a curved path, as opposed to the straight-line distance between two points. It appears throughout mathematics, physics, and engineering, from the length of a road on a map, to the path of a satellite orbit, to the amount of cable needed to follow a curved route. For circles, arc length has an elegant exact formula. For arbitrary curves, it requires integration.
The Circle Arc Formula
The radian is defined so that one radian corresponds to an arc equal in length to the radius. This makes the circle arc formula beautifully simple:
| Angle | Arc length (r = 1) | Sector area (r = 1) |
|---|---|---|
| 30° (π/6) | 0.5236 | 0.2618 |
| 45° (π/4) | 0.7854 | 0.3927 |
| 90° (π/2) | 1.5708 | 0.7854 |
| 180° (π) | 3.1416 | 1.5708 |
| 360° (2π) | 6.2832 | 3.1416 |
The Curve Arc Length Formula, Where It Comes From
For a smooth curve y = f(x) from x = a to x = b, imagine the curve divided into infinitely many tiny straight segments. Each segment has horizontal width dx and vertical rise dy = f′(x)·dx. By the Pythagorean theorem, the length of each segment is:
Summing all these tiny segments, integrating from a to b, gives the arc length formula. This integral is often difficult or impossible to evaluate analytically, which is why numerical methods like Simpson's rule are used for arbitrary functions.
Real-World Applications of Arc Length
- ▸Road and rail engineering: Calculating the length of curved road or track sections for material estimates and speed limits.
- ▸Surveying: Measuring the length of boundary curves or rivers on topographic maps.
- ▸Manufacturing: Determining how much wire, cable, pipe, or material is needed to follow a curved path.
- ▸Physics: Computing the path length of a projectile, the length of a pendulum arc, or a satellite orbit segment.
- ▸Computer graphics: Parameterising curves by arc length for uniform-speed animation and smooth camera paths.
- ▸Optics: Calculating the path length of light through curved lenses and mirrors.
Common Arc Length Results
| Curve | Interval | Arc Length |
|---|---|---|
| y = x | [0, 1] | √2 ≈ 1.4142 |
| y = x² | [0, 1] | ≈ 1.4789 |
| y = x² | [0, 2] | ≈ 4.6468 |
| y = sin(x) | [0, π] | ≈ 3.8202 |
| y = √x | [0, 4] | ≈ 4.6478 |
| y = ln(x) | [1, e] | ≈ 2.0035 |
| Circle arc | r=1, θ=2π | 2π ≈ 6.2832 (full circle) |
Frequently Asked Questions
What is arc length?
Arc length is the actual distance along a curve between two points, not the straight-line (chord) distance. If you think of a curved road between two cities, arc length is how far you drive, while chord length is how far apart the cities are on a straight line.
Why does the arc length formula contain √(1 + [f′(x)]²)?
The formula comes from infinitesimal geometry. A tiny segment of a curve has horizontal run dx and vertical rise dy = f′(x) dx. By the Pythagorean theorem, the length of that segment is ds = √(dx² + dy²) = √(1 + [f′(x)]²) dx. Integrating ds over [a,b] gives the total arc length.
What functions can I enter?
You can use standard mathematical notation: x^2 (x²), x^(1/2) or sqrt(x), sin(x), cos(x), tan(x), log(x) (natural log), log10(x), exp(x) (eˣ), abs(x), and combinations. Multiplication must be explicit: 2*x, not 2x.
How accurate is the numerical integration?
Simpson's rule with 1000 subintervals is accurate to approximately 6–8 significant figures for smooth functions. For functions with sharp corners or near-discontinuities, accuracy may decrease. The central difference derivative approximation introduces an additional error of order h² ≈ 10⁻¹⁰.
What is the difference between arc length and chord length?
Arc length is the curved path distance; chord length is the straight-line distance between the two endpoints. For a curve y = f(x) from a to b, chord length = √((b−a)² + (f(b)−f(a))²). Arc length is always ≥ chord length, with equality only for a straight line.
What is a circle sector and how does it relate to arc length?
A circle sector is a "pie slice", a region bounded by two radii and the arc between them. The arc length of the sector boundary is L = rθ. The area of the sector is A = ½r²θ. The chord (straight line connecting the two arc endpoints) has length 2r sin(θ/2).
Can I calculate arc length in 3D?
This calculator handles 2D curves only. For 3D space curves r(t) = (x(t), y(t), z(t)), the arc length formula is L = ∫|r′(t)| dt = ∫√(x′² + y′² + z′²) dt over the parameter range. This requires a parametric arc length calculator.