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.
Description
A full integer-ratio epicycloid encloses an area determined by its rolling radius and radius ratio.
A full integer-ratio epicycloid encloses an area determined by its rolling radius and radius ratio. The geometric convention differs from an internally rolling hypocycloid.1
Exact full simple closed integer-ratio epicycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
When to use Epicycloid Area Calculator
Use this tool to exact full simple closed integer-ratio epicycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Epicycloid 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 rollingRadius, fixedToRollingRadiusIntegerRatio using the field table and calculation rules, then run the tool.
- Read curveMeasure. 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 |
|---|---|---|---|
| rollingRadius | Rolling radius | number | Required |
| fixedToRollingRadiusIntegerRatio | Fixed to rolling radius integer ratio | number | Required |
Enter positive rolling radius r and integer fixed/rolling ratio k in [1,one million]. Evaluate full enclosed area in squared radius units.
area = πr²(k+1)(k+2) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| curveMeasure | Curve measure | number |
{
"input": {
"rollingRadius": 1,
"fixedToRollingRadiusIntegerRatio": 3
},
"output": {
"curveMeasure": 62.83185307179586
}
}Limitations and assumptions
Only the stated simple closed integer-ratio curve is included.
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 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. For a complementary task, use Hypocycloid Area Calculator: Exact full simple closed integer-ratio hypocycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
References
-
Epicycloid Area Calculator — tool.io
Similar or alternative tools
- 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.
- Hypocycloid Area Calculator
Exact full simple closed integer-ratio hypocycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.