Arc Length Calculator
Calculate the arc length of linear, quadratic, and circular curves between two points using numerical integration.
About this calculator
This calculator measures the length of a curve — linear, quadratic, or a circular arc, chosen via Curve Type — between x Start and x End, by numerically integrating L = ∫ √(1 + f′(x)²) dx with Simpson's rule across 200 subintervals (n = 200). It also reports the straight-line Chord Length between the two endpoints and the Arc/Chord Ratio, which the engine defines as exactly 1 by a fallback (chordLength > 0 ? ... : 1) whenever the chord collapses to zero rather than leaving the ratio undefined. Among the three numeric inputs, x End has the largest measured effect on Arc Length at the calculator's default quadratic curve, ahead of Coefficient a; x Start and Coefficient b are both left out of that ranking entirely because they default to 0, and the sensitivity probe scales an input by a percentage of itself — a percentage of zero is still zero, so a 0-valued input can never register a measurable effect at all.
Curve Type itself moves Arc Length more than any single numeric input does, though, since switching between the linear, quadratic, and circular formulas changes which curve is being measured entirely rather than nudging a parameter within one curve. What the calculator does NOT account for: setting x End below x Start makes the integration step size negative, and Simpson's rule then sums those negative-width slices directly, so the reported Arc Length itself comes back as a negative number rather than the positive distance the name implies.
Inputs
Results
Arc Length
4.64678
How to Use This Calculator
- Select the curve type (linear, quadratic, or circular) and enter its defining parameters.
- Enter x Start and x End — the lower and upper bounds along the x-axis over which to measure the arc length.
- Arc length L = ∫[a to b] √(1 + (f'(x))²) dx; the calculator evaluates this integral numerically.
- For all curve types, including the circle, arc length is evaluated numerically with Simpson's rule across 200 subintervals — not via a closed-form formula.
- The curve chart shows the segment of the function between the two bounds so you can visually verify the range.
- Arc length is always ≥ the straight-line distance between the endpoints (chord length).
How the result changes with x End
| x End | Arc Length |
|---|---|
| 1 | 1.47894 |
| 1.5 | 2.82632 |
| 3 | 9.74709 |
| 5 | 25.87424 |
What each input means
- Curve Type
- Type of curve to measure
- Coefficient a
- Leading coefficient (or radius for circle)
- Coefficient b
- Second coefficient (not used for circle)
- x Start
- Starting x value
- x End
- Ending x value
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersCurve Type = 2, Coefficient a = 1, Coefficient b = 0, x Start = 0 = 5 input(s) provided
- Calculate Arc LengthArc Length4.64678 = 4.64678
- Calculate Chord LengthChord Length4.47214 = 4.47214
- Calculate Arc/Chord RatioArc/Chord Ratio1.0391 = 1.0391
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why doesn't x Start ever show up as affecting the Arc Length in a sensitivity check?
x Start defaults to 0, and the sensitivity probe tests each input by scaling it up or down by a percentage of its own value. A percentage of zero is still zero, so the probe never actually moves x Start away from 0 at all — it isn't that x Start has no effect, it's that the probe can't test an input starting at exactly zero.
Can the Arc Length come out negative?
Yes, if x End is set lower than x Start. The integration step size is computed as (xEnd - xStart) / n, which goes negative when xEnd is smaller, and Simpson's rule then sums negative-width slices across the whole curve, producing a negative total — even though a physical arc length should never be less than zero.
What does the Arc/Chord Ratio equal when the two endpoints coincide?
The engine hardcodes it to exactly 1 in that case: chordLength > 0 ? Math.round((arcLength / chordLength) * 10000) / 10000 : 1. Rather than leaving the ratio undefined or throwing a division error when the chord length is zero, it substitutes the value a straight, zero-length segment would trivially satisfy.
Why does increasing Coefficient a increase the Arc Length for the default quadratic curve?
For the default parabola f(x) = a·x², the derivative f′(x) = 2ax grows directly with a, and a steeper derivative makes the integrand √(1 + f′(x)²) larger at every point along the curve. A larger Coefficient a therefore produces a visibly steeper, longer curve between the same two x bounds, which the Arc Length calculation reflects directly.
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