Cycloid Arc Length Calculator
Compute exact full ordinary cycloid arch length (eight radii per arch); does not apply to incomplete or altered trochoid trajectories.
Description
A full ordinary cycloid arch has length eight times its generating radius.
A full ordinary cycloid arch has length eight times its generating radius. Counting complete arches avoids assumptions about a partial trajectory's endpoints.1
Compute exact full ordinary cycloid arch length (eight radii per arch); does not apply to incomplete or altered trochoid trajectories.
When to use Cycloid Arc Length Calculator
Use this tool to compute exact full ordinary cycloid arch length (eight radii per arch); does not apply to incomplete or altered trochoid trajectories.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Cycloid Arc Length Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter rollingCircleRadius, completeArchCount using the field table and calculation rules, then run the tool.
- Read completeCycloidArcLength. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| rollingCircleRadius | Rolling circle radius | number | Required |
| completeArchCount | Complete arch count | number | Required |
Supply positive rolling radius and integer whole-arch count 0–one million. Return complete-arch length in radius units.
length = 8r × complete arch count Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| completeCycloidArcLength | Complete cycloid arc length | number |
{
"input": {
"rollingCircleRadius": 2,
"completeArchCount": 3
},
"output": {
"completeCycloidArcLength": 48
}
}Limitations and assumptions
The relation does not describe incomplete or altered trochoid trajectories.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Cycloid Area Calculator: Compute planar area between baseline and whole ordinary cycloid arches from rolling-circle radius; not curtate/prolate trochoids or arbitrary incomplete arcs. For a complementary task, use Epicycloid Perimeter Calculator: Exact full simple closed integer-ratio epicycloid perimeter; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
References
-
Cycloid Arc Length Calculator — tool.io
Similar or alternative tools
- Cycloid Area Calculator
Compute planar area between baseline and whole ordinary cycloid arches from rolling-circle radius; not curtate/prolate trochoids or arbitrary incomplete arcs.
- Epicycloid Perimeter Calculator
Exact full simple closed integer-ratio epicycloid perimeter; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.