Length is not signed area
The square-root integrand is a speed and is nonnegative. Traversing a curve backward does not make its geometric length negative.
Arc Length Calculator
Find the length of a graph y=f(x) or a planar parametric curve x(t), y(t). The calculator differentiates your expressions, builds the arc-length integral, evaluates it numerically, and graphs the traced curve—all locally in your browser.
Enter a curve
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Arc length result
Calculated curve length
—
Enter a curve and bounds to calculate its length.
The arc-length integral will appear here.
Calculated derivatives will appear here.
The result is a numerical approximation for a smooth curve on the selected interval.
Curve graph
Calculate a curve to draw the traced segment.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Parse the curve → differentiate symbolically → form the speed integrand → scan the interval for non-real or extreme values → integrate with adaptive Simpson subdivision.
- Accuracy
- The result includes the quadrature error estimate produced by the final subdivisions. It is not a proof of the exact rounding error.
- Input limits
- Real-valued expressions of at most 300 characters and finite bounds between −10¹² and 10¹². The curve must be smooth throughout the interval.
- Privacy
- Your equations, bounds, graph, copied summary, and downloaded samples remain on your device.
How to use the arc length calculator
- Choose Function for a graph
y=f(x), or Parametric for coordinatesx(t)andy(t). - Enter the expression or expressions using
^for powers. Trigonometric inputs use radians. - Enter the finite interval. Reversed bounds are allowed and produce the same nonnegative geometric length.
- Calculate, then inspect the derivatives, integral, error estimate, and curve graph. Copy the summary or download the plotted samples as CSV if useful.
How to interpret the result
Function mode
For y=f(x), each small horizontal change dx combines with the vertical change f′(x)dx through the Pythagorean theorem.
Parametric mode
The vector velocity is (x′(t),y′(t)). Its magnitude is integrated over the parameter interval, so the parameter itself need not measure distance.
Corners and singularities
The numerical method expects smooth input. Split a piecewise-smooth curve at corners, cusps, or singular points, calculate each smooth segment, and add the lengths.
Worked examples
Line y=x on [0,1]
f′(x)=1, so L=∫₀¹√(1+1²)dx=√2≈1.41421356237.
Unit circle
For x(t)=cos(t) and y(t)=sin(t), speed is √(sin²t+cos²t)=1. From 0 to 2π, the length is 2π≈6.28318530718.
Arc length formulas
| Curve form | Conditions | Length |
|---|---|---|
Function y=f(x) | f′ continuous on [a,b] | L=∫ₐᵇ √(1+[f′(x)]²) dx |
Parametric x=x(t), y=y(t) | x′ and y′ continuous on [a,b] | L=∫ₐᵇ √([x′(t)]²+[y′(t)]²) dt |
These formulas apply directly to continuously differentiable curves. Piecewise-smooth curves can be handled segment by segment. A curve that retraces the same geometric segment counts the distance each time it is traversed.
Reference: the standard derivations and conditions are described in OpenStax Calculus Volume 2, §2.4 and Calculus Volume 2, §7.2.
Arc length calculator FAQs
What is the arc length formula for y=f(x)?
For a continuously differentiable function on [a,b], use L=∫ₐᵇ√(1+[f′(x)]²)dx. The 1 represents the horizontal component of each small displacement.
How do I calculate the length of a parametric curve?
Differentiate both coordinates with respect to the same parameter, find the speed √([x′(t)]²+[y′(t)]²), and integrate that speed across the parameter interval.
Why is the result still positive when I reverse the bounds?
Arc length is distance, not an oriented integral. Reversing the bounds changes the direction of traversal but not the distance traveled, so this calculator orders the bounds before integrating.
Does a parametric curve that retraces itself count twice?
Yes. The speed integral measures distance traveled along the parameterization. If the parameterization traces a segment twice, both traversals contribute to the result.
Can I use degrees for sine and cosine?
No. Calculus derivatives for trigonometric functions assume radians. Convert degrees to radians before entering a parameter interval.
What if the curve has a corner or cusp?
Split the interval at each nonsmooth point, calculate the smooth pieces separately, and add their nonnegative lengths. The calculator reports derivatives that are undefined or extreme inside a single interval.
Is the numerical error estimate guaranteed?
No. It estimates the remaining adaptive Simpson quadrature error based on differences between successive subdivisions. It can be less reliable for badly behaved or insufficiently resolved curves, which is why the calculator also scans for non-real values and extreme derivatives.
Does my curve data leave the browser?
No. All parsing, differentiation, integration, graphing, copying, and file preparation happen locally. A permalink contains only the inputs you explicitly choose to put in the URL.