Arc Length Of A Curve Calculator
Work out arc length of a curve instantly with clear inputs, formula shown and shareable results.
About Arc Length Of A Curve Calculator
Arc length integrates the hypotenuse of each infinitesimal step: L = ∫√(1 + (f'(x))²) dx. The chord is always shorter, and the difference measures how much the curve bends.
Formula
Arc length
L = ∫ₐᵇ √(1 + (f'(x))²) dx
Worked example
With the default values below, Arc Length Of A Curve Calculator returns Arc length of 3.16784090. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Curve coefficients, highest power first
- 1, 0, 0
- From x =
- 1
- To x =
- 2
Results
- Arc length
- 3.16784090
- Straight-line chord
- 3.16227766
- Extra length from curvature
- 0.00556324
- Horizontal span
- 1.000000
Step by step
- 1Values used
- 2Arc length
- 3Arc length= 3.16784090
- 4Straight-line chord= 3.16227766
- 5Extra length from curvature= 0.00556324
- 6Horizontal span= 1.000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Arc length of y = x² from 0 to 1?
About 1.4789428, slightly more than the chord √2 ≈ 1.4142136.
Why are arc lengths usually numerical?
The square root of a polynomial rarely has an elementary antiderivative, so quadrature is the practical route.