Arc Length Calculator
Find the exact length of a curve y = f(x) over an interval [a, b] using numerical integration.
Function f(x)
Length of y = x^2 from x = 0 to x = 2
How does this calculator work?
Arc length measures the true distance along a curve y = f(x) from x = a to x = b, computed as the integral of √(1 + f'(x)²). This tool differentiates numerically with central differences and integrates with Simpson's rule (n = 1000) to return an accurate length plus a plot of the curve.
Formula
How this is calculated
The arc length of a smooth curve y = f(x) from x = a to x = b equals the integral of √(1 + f'(x)²). Pick the function from the list and enter the lower limit a and upper limit b; the interval may be entered in either order and the calculator integrates over [min(a,b), max(a,b)].
The derivative f'(x) is approximated by a central finite difference, f'(x) ≈ (f(x+h) − f(x−h)) / (2h) with h = 1e-6, which is accurate for the smooth functions provided. The integral is then evaluated with the composite Simpson's rule using n = 1000 subintervals, giving a tight numerical estimate of the true arc length.
Results are in the same length units as the x and y axes (they must share the same unit). Choose limits inside each function's domain: for example sqrt(x) and ln(x) require x > 0, and 1/x must not straddle x = 0. The chord value shows the straight-line distance between the endpoints, which is always less than or equal to the arc length.
Frequently asked questions
The chord is the shortest path between the two endpoints (a straight line), while the arc follows every bend of the curve. Unless the curve is itself a straight line, the arc length is always greater than the chord.
It uses composite Simpson's rule with 1000 subintervals and a central-difference derivative, so for the smooth functions offered the error is typically far below the four decimal places shown.
Arc length carries the same unit as the x and y axes, which must be identical. If x and y are both in meters, the length is in meters.
Also known as
TG we-Calculate Editorial Team. (2026). Arc Length Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/arc-length-calculator
TG we-Calculate Editorial Team. "Arc Length Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/arc-length-calculator.
TG we-Calculate Editorial Team, "Arc Length Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/arc-length-calculator
@misc{wecalculate_arc_length_calculator, title = {Arc Length Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/arc-length-calculator}}, year = {2026}, note = {TG we-Calculate} }
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