Calculator Inputs
Use explicit multiplication in formulas, such as 2*x, 3*sin(x), or sqrt(1-x^2).
Example Data Table
These examples show common arc length scenarios across all supported input styles.
| Mode | Sample inputs | Expected interpretation | Approximate output |
|---|---|---|---|
| Circle arc | Radius = 10, Angle = 90 degrees | Quarter-circle path | 15.7079632679 |
| Function y(x) | y = sqrt(1-x^2), x from -1 to 1 | Upper semicircle of radius 1 | 3.1415926536 |
| Parametric | x = 3*cos(t), y = 2*sin(t), t from 0 to 2π | Ellipse perimeter approximation | 15.8654395893 |
| Polar | r = 2 + cos(θ), θ from 0 to 360 degrees | Closed limacon arc | 13.3648932206 |
Formula Used
1) Circle arc: s = rθ, where θ must be in radians.
2) Cartesian function: L = ∫ from a to b √(1 + (dy/dx)^2) dx.
3) Parametric curve: L = ∫ from a to b √((dx/dt)^2 + (dy/dt)^2) dt.
4) Polar curve: L = ∫ from a to b √(r^2 + (dr/dθ)^2) dθ.
This calculator uses Simpson’s Rule for numerical integration in function, parametric, and polar modes. Derivatives are estimated numerically with adaptive finite differences.
How to Use This Calculator
- Choose the input mode that matches your geometry: circle, function, parametric, or polar.
- Enter the required values or formulas. Use explicit multiplication, such as
2*x. - Select angle units where relevant. Circle and polar modes accept degrees or radians.
- Set integration intervals for advanced modes. Larger values usually improve numeric smoothness.
- Click Calculate Arc Length. The results appear immediately below the header and above the form.
- Review summary metrics, detailed outputs, and the plotted path.
- Use the export buttons to save the result as CSV or PDF for documentation.
Frequently Asked Questions
1) What is arc length?
Arc length is the distance measured along a curve between two points. It follows the actual path, not a straight shortcut between endpoints.
2) Why does the circle formula require radians?
The compact formula s = rθ only works directly when the angle is expressed in radians. Degrees are converted automatically before calculation.
3) What expressions are supported?
You can use numbers, parentheses, powers, and common functions like sin, cos, tan, sqrt, abs, exp, log, and log10.
4) How accurate are the advanced modes?
They are usually very accurate for smooth curves. Increasing the integration intervals can improve stability, especially for long or sharply changing paths.
5) Why might a formula fail to evaluate?
An expression may fail if it contains unsupported symbols, missing multiplication signs, invalid domains, or values that produce undefined results such as square roots of negatives.
6) What is the difference between chord length and arc length?
Chord length is the straight-line distance between endpoints. Arc length follows the curve itself, so it is always at least as large as the chord.
7) Can I use this for engineering or CAD checks?
Yes, it is useful for verification, estimation, and study. For mission-critical work, compare results against a domain-specific tool or analytical reference.
8) What does the graph help me verify?
The graph helps confirm the selected path, endpoint positions, overall curve shape, and whether the entered formulas behave as expected over the chosen interval.