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.
Description
An ordinary cycloid is generated by a circle rolling along a straight baseline.
An ordinary cycloid is generated by a circle rolling along a straight baseline. Whole arches have a fixed area relative to the generating circle.1
Compute planar area between baseline and whole ordinary cycloid arches from rolling-circle radius; not curtate/prolate trochoids or arbitrary incomplete arcs.
When to use Cycloid Area Calculator
Use this tool to compute planar area between baseline and whole ordinary cycloid arches from rolling-circle radius; not curtate/prolate trochoids or arbitrary incomplete arcs.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Cycloid Area 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 areaUnderCompleteCycloidArches. 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 |
Enter positive rolling radius and integer complete-arch count from 0 to one million. Return area under those arches in squared radius units; zero arches gives zero.
area = 3πr² × complete arch count Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| areaUnderCompleteCycloidArches | Area under complete cycloid arches | number |
{
"input": {
"rollingCircleRadius": 2,
"completeArchCount": 3
},
"output": {
"areaUnderCompleteCycloidArches": 113.09733552923255
}
}Limitations and assumptions
Incomplete arcs and curtate/prolate trochoids do not use this formula.
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 Arc Length Calculator: Compute exact full ordinary cycloid arch length (eight radii per arch); does not apply to incomplete or altered trochoid trajectories. For a complementary task, use Epicycloid Area Calculator: Exact full simple closed integer-ratio epicycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
References
-
Cycloid Area Calculator — tool.io
Similar or alternative tools
- Cycloid Arc Length Calculator
Compute exact full ordinary cycloid arch length (eight radii per arch); does not apply to incomplete or altered trochoid trajectories.
- Epicycloid Area Calculator
Exact full simple closed integer-ratio epicycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.