Arc Length of a Function Calculator
Approximate arc length from sampled points on a function curve. This educational calculator shows the formula, result, and step-by-step interpretation.
Calculator
What this calculator teaches
Arc length measures distance along a curve and connects geometry with integration.
Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.
From Sampled Points to Arc-Length Approximation
When a curve is represented by sampled coordinates rather than an explicit formula, its length can be estimated by joining consecutive points with straight segments. For points (xᵢ,yᵢ), each segment has length √[(Δx)²+(Δy)²], and the polyline length is the sum of those segment lengths.
This is a discrete approximation to the continuous arc-length integral. For a differentiable function y=f(x), the exact theoretical form is L=∫√[1+(f′(x))²]dx over the interval. With samples, the calculator does not infer f′; it simply measures the polygonal path through the points you provide.
Accuracy depends strongly on sampling. Points should follow the curve in order, and more points are especially valuable where curvature is high. Adding samples to a smooth curve generally makes the polyline track the curve more closely, although noisy measurements can make the measured path longer rather than more accurate.
| Quantity | Rule |
|---|---|
| Segment length | dᵢ=√[(xᵢ−xᵢ₋₁)²+(yᵢ−yᵢ₋₁)²] |
| Polyline length | L≈Σdᵢ |
Because Euclidean distance is used, x and y should be expressed in compatible units. Mixing meters on one axis with centimeters on the other without conversion changes the geometry.
Sanity-check the path length against straight-line distance. The summed path through several points can never be shorter than the direct Euclidean distance between the first and last points. Equality occurs when all intermediate points lie on the straight segment in order. If your computed path is shorter, the coordinates or arithmetic are inconsistent.
Sampling density should follow geometry rather than a fixed habit. Nearly straight portions need fewer points, while tight bends need more. For measured GPS or sensor data, oversampling can also capture small noise and inflate the path length, so smoothing and measurement resolution may matter as much as the distance formula itself.
Formula & Symbols
| Concept | Formula or rule |
|---|---|
| Polyline approximation | Σ sqrt((Δx)² + (Δy)²) |
Worked example
Common mistakes
Keep lists comma separated, matrices as rows separated by semicolons, and modular inputs as integers.
Some methods require positive probabilities, valid moduli, independent trials, or small educational input sizes.
The calculator connects points exactly in the listed sequence. Reordering the same coordinates can create a different path and a different length.
Unless the underlying curve is itself piecewise linear, a finite set of straight segments is an approximation.
FAQ
Related calculators
These links will work after the calculators are registered in the final Math layout update.
Formula Explorer connections
Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.