Engineering Calc Tools

Calculus

Arc Length Calculator

Length of a curve y = f(x) between two points, with the derivative and a chord check.

Tool no.
ECT-C05
Rev
1.0
Updated
Oct 2026
Method
Symbolic derivative + adaptive Gauss–Kronrod
Examples

Given

Use x as the variable. Supports + − * / ^, sqrt, exp, ln, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, abs, pi and e.

Limits accept expressions such as pi/2 or 2*e.

The curve is also approximated by n straight segments, which are drawn on the graph.

Solution

Arc length —
Runs in your browser. Nothing you enter is uploaded.Symbolic derivative · adaptive Gauss–Kronrod 7/15
How to use this calculator

Type the curve as a function of x and the x-values where the piece you want to measure starts and ends. The large number is the arc length. The calculator differentiates f symbolically, builds the integrand √(1 + f′²), and integrates it with adaptive Gauss–Kronrod quadrature to about 12 significant figures. As a check it also adds up n straight chords between points on the curve, which are drawn on the graph.

Formula

Arc length of y = f(x) from a to bL = ∫ab √(1 + [f′(x)]²) dx
Where it comes fromds = √(dx² + dy²) = √(1 + (dy/dx)²) dx

Zoom in far enough and any smooth curve looks straight. A tiny piece with horizontal run dx and rise dy = f′(x) dx has length √(dx² + dy²) by the Pythagorean theorem. Adding up all the pieces gives the integral.

Worked example

The parabola y = x² from x = 0 to 1 (the default)

  1. f′(x) = 2x, so the integrand is √(1 + 4x²).
  2. With the substitution 2x = sinh u, the exact answer is (2√5 + sinh⁻¹ 2)/4.
  3. L = 1.478942857544.
  4. Four chords give 1.474281, about 0.3% short, because chords always cut corners.

Curves with exact answers

Only a few curves have arc lengths in closed form, which is why numerical integration is the practical tool. Good ones to check your work against: a straight line (the distance formula), y = cosh x (L = sinh b − sinh a, the hanging-chain catenary), y = x3/2 (L = (8/27)[(1 + 9b/4)3/2 − 1] from 0), and a circle (L = rθ). The simple-looking y = sin x already needs an elliptic integral: its length over one hump is 3.8202.

Arc length in engineering

Arc length gives the amount of material in a curved part: the length of a cable sagging between towers, of a bent pipe or rebar, or of a cam profile. It also turns a path into a distance, as in the length of a highway curve or a robot’s tool path.

Frequently asked questions

Why does the calculator fail on a full semicircle?

At x = ±1 the semicircle √(1 − x²) has a vertical tangent, so f′ is infinite there. Use limits slightly inside, such as −0.999 to 0.999, or use the circle formula L = rθ.

Does the direction of the limits matter?

No. Arc length is a distance, so it is always positive. If you enter b < a the calculator measures the same piece of curve.

What about parametric or polar curves?

Those use L = ∫√(x′² + y′²) dt and L = ∫√(r² + r′²) dθ. This calculator handles curves written as y = f(x).