Limacon Area Calculator

Compute one 0–2π polar trace area for r=a+b cos θ; when b>a inner/outer loops are both counted by polar integral, not a geometric union area.

Description

A limaçon r=a+b cosθ can include an inner loop when b>a.

A limaçon r=a+b cosθ can include an inner loop when b>a. A polar area integral over one full trace counts loop contributions according to the trace, rather than taking a geometric union.^[1]

Compute one 0–2π polar trace area for r=a+b cos θ; when b>a inner/outer loops are both counted by polar integral, not a geometric union area.

When to use Limacon Area Calculator

Use this tool to compute one 0–2π polar trace area for r=a+b cos θ; when b>a inner/outer loops are both counted by polar integral, not a geometric union area.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Limacon 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 radialOffset, cosineAmplitude using the field table and calculation rules, then run the tool.
Read tracedPolarArea, hasInnerLoop. 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
radialOffset	Radial offset	number	Required
cosineAmplitude	Cosine amplitude	number	Required

Use nonnegative radial offset and cosine amplitude with at least one nonzero. Return the 0–2π traced polar area and a flag for amplitude strictly greater than offset.

traced area = π(a²+b²/2)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
tracedPolarArea	Traced polar area	number
hasInnerLoop	Has inner loop	boolean

json

{
  "input": {
    "radialOffset": 2,
    "cosineAmplitude": 1
  },
  "output": {
    "tracedPolarArea": 14.137166941154069,
    "hasInnerLoop": false
  }
}

Limitations and assumptions

The returned area is not the union area of overlapping loop interiors.

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. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Rose Area Calculator: Compute distinct rose petals for r=a cos(kθ), integer k; odd k repeats the curve over 2π, so repeated traces are not counted as extra area. For a complementary task, use Bernoulli Lemniscate Area Calculator: Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.

References

Limacon Area Calculator — tool.io

Similar or alternative tools

Rose Area Calculator
Compute distinct rose petals for r=a cos(kθ), integer k; odd k repeats the curve over 2π, so repeated traces are not counted as extra area.
Bernoulli Lemniscate Area Calculator
Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.