Arc Length Calculator
Compute curve length for explicit, parametric, polar or vector-valued functions using derivatives and numerical integration across a specified interval precisely.
Inputs
Equation Preview
Helping Notes
Use radians for trigonometric inputs; write powers with ^ and functions like sin(x), cos(x), exp(x), ln(x).
Arc length formulas integrate a speed term over the interval; ensure the function is differentiable on that interval.
Results
Inputs Summary
Arc Length
Computation Details
What Is an Arc Length Calculator?
An Arc Length Calculator determines the distance along a curve between two points. It supports multiple representations: explicit functions , inverse functions , parametric curves in 2D, and general 3D paths . You choose an interval, the tool sets up the correct integral, evaluates symbolically when possible, and otherwise performs accurate numeric approximation. It also reports intermediate steps, derivatives, and unit‑aware results, so you can verify method and copy formulas into notes. Applications include geometry, physics, CAD, animation pathing, surveying, and engineering design where precise path lengths are essential.
About the Arc Length Calculator
The core idea is to integrate the speed of traversal along the curve. For explicit functions, speed comes from the Pythagorean combination of and , producing the well‑known square‑root integrand. Parametric and polar forms follow the same principle, adapting to the chosen variables. Many arc‑length integrals lack elementary antiderivatives; the calculator detects such cases and switches to robust numerical methods (adaptive quadrature, Simpson/trapezoid sums) with controllable tolerance. Piecewise curves are handled by summing sub‑interval lengths. Outputs include exact forms where available, high‑precision numerics, and optional reparameterization by arc length for downstream computations like constant‑speed traversal.
Explicit function:
Inverse form:
Parametric (2D):
Parametric (3D):
Polar:
Riemann/Simpson approximation:
How to Use This Arc Length Calculator
- Choose representation: function, parametric, polar, or 3D. Enter the curve and the interval (or ).
- Compute derivatives automatically (or paste your own). The tool builds the correct integrand for length.
- Click calculate. You’ll get exact results when available and a high‑precision numeric otherwise.
- Review steps, units, and (optionally) enable a numerical check using partition sums for validation.
Examples
- Circle (parametric): ⇒ , .
- Parabola segment: ⇒ .
- Polar spiral: ⇒ .
- 3D helix: ⇒ , .
Formula Snippets Ready for Rendering
FAQs
Does the Arc Length Calculator support parametric curves?
Yes. Provide x(t), y(t) (and z(t) for 3D); the calculator integrates the speed over your parameter interval.
Can it handle polar equations?
Yes. Use the polar formula with angles in radians for correct results.
What if the integral has no elementary antiderivative?
The tool switches to accurate numerical integration and shows the method used and precision target.
How are units treated?
Input units carry through unchanged—centimeters in, centimeters out—since length is a geometric measure.
Can I sum piecewise lengths?
Yes. The total is the sum of lengths over sub‑intervals; the calculator automates the partitioning.
What accuracy can I expect from numerics?
Choose a tolerance; adaptive methods refine partitions until the estimated error falls below your target.
Does it support curves with cusps or corners?
Points where derivatives blow up are handled by splitting intervals; the tool flags problematic points with guidance.
Why are radians recommended for polar inputs?
Most formulas are derived in radians; using degrees without conversion yields incorrect arc lengths.
Can I reparameterize by arc length?
Yes. The calculator can produce an s(t) mapping and approximate its inverse for constant‑speed traversal.
How do I validate a numeric answer?
Compare against a Riemann‑sum estimate or differentiate a known closed form to check consistency.
What about self‑intersecting curves?
Arc length is traced over the parameter interval; intersections don’t cancel—length accumulates along the full path.