ใฐ๏ธArc Length Calculator
Find the arc length of a curve y=f(x) or a parametric curve, with step-by-step setup and a highlighted graph.
Calculations run entirely in your browser using math.js for derivatives and Simpson's rule for the integral. No data is sent to a server.
Free Arc Length Calculator
Need the length of a curve between two points, not just its equation? Our free arc length calculator applies the arc length integral instantly and shows the full setup. Choose an explicit function y = f(x) or a parametric curve x(t), y(t), enter your interval, and see the derivative, the integrand, and the final length computed step by step.
When the resulting integral has no elementary antiderivative โ which is common for arc length problems โ the calculator uses numerical integration (Simpson's rule) and clearly labels the result, so you always get an answer. A graph highlights the exact segment of the curve being measured, making it easy to sanity-check the result visually.
Enter your curve and interval to get started. No sign-up, no downloads, no limits โ just clear, worked-out calculus whenever you need it. See the About page or the FAQ for more.
About the Arc Length Calculator
Features
Explicit & parametric modes
Handles y=f(x) curves and x(t), y(t) parametric paths.
Step-by-step setup
Derivative and integrand shown before the final number.
Numerical evaluation
Simpson's rule with 2000 subintervals for high accuracy.
Highlighted graph
The measured segment is drawn over the full curve.
Frequently Asked Questions
How does the arc length calculator work?
Choose explicit or parametric mode, enter your curve and interval, and it applies the arc length integral instantly with steps shown.
Is it free?
Completely free, no sign-up, downloads, or limits.
Does it work for parametric curves?
Yes, switch to parametric mode and enter x(t) and y(t) with the range of t.
What if the integral has no elementary antiderivative?
It falls back to numerical integration and clearly labels the result as an approximation.
Does it show the setup, not just the final number?
Yes, the derivative, the integrand, and the bound substitution are all shown.
What happens with a vertical tangent in the interval?
It's detected and explained clearly rather than producing an error or wrong value.
Does it graph the curve?
Yes, with the measured segment between the bounds highlighted.
Are results exact or decimal?
Clean results are shown as exact values; otherwise decimals are used.
Is my input stored anywhere?
No; everything runs in your browser and isn't stored or sent anywhere.
Does it work on mobile?
Yes, fully responsive, including the graph and step-by-step panel.