Limacon Area Calculator
A limaçon is a polar curve defined by r = a + b·cos θ (or the sine form). Depending on the two constants a and b, it can be a heart-shaped cardioid, a dimpled curve, a convex oval, or a looped figure. Enter a and b below to get the area from the closed-form polar area formula, the curve type detected automatically, the polar equation you actually computed, and the a/b ratio that decides the shape.
Use the equation-type selector to switch between the cosine and sine forms. The enclosed area is identical for both; only the orientation of the curve changes.
For study and homework use. Enter positive values for a and b.
Quick Answer
For the pre-filled example, r = 3 + 2·cos θ, the enclosed area is 34.5575 square units. The tool classifies the curve as dimpled (3 is larger than 2 but smaller than twice 2), prints the equation r = 3 + 2 cos θ, and reports an a/b ratio of 1.500. The formula line shows the arithmetic: π(9 + 2) = 34.5575.
Key takeaways
- One formula covers every limaçon: A = π(a² + b²/2). It is the polar area integral ½∫r² dθ over a full turn, with the cross term integrating to zero.
- The a/b ratio is the whole classification: exactly 1 is a cardioid, between 1 and 2 is dimpled, 2 or more is convex, below 1 grows an inner loop. The tool applies the exact values, not the rounded ratio it prints.
- For an inner-loop curve the printed area is the full integral, inner loop counted on top of the outer region. The inner-loop section below gives the verified split for r = 1 + 2 cos θ so you can see what the single number contains.
- Quick Answer
- How to use the limacon area calculator
- The formula: π(a² + b²/2), and where it comes from
- The four limacon shapes
- Worked examples, checked against this exact tool
- Cosine vs sine form
- The inner-loop case, stated precisely
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
How to use the limacon area calculator
Read the two constants off your polar equation, pick the cosine or sine form, and click Calculate. The result stays hidden until you click, and Reset restores the worked example.
- Find a and b: in r = a + b·cos θ, a is the constant term and b is the number multiplying the trig term. In r = 3 + 2 cos θ, a = 3 and b = 2. If your equation is written the other way around, as r = 2 + 3 cos θ, then a = 2 and b = 3, and the shape changes with the swap.
- Pick the equation form: cosine or sine, matching your problem statement. The area is the same either way; the printed equation line keeps the form you chose, which prevents copying errors in homework.
- Both values must be positive: the tool rejects zero and negative entries with a specific message (“Parameter a must be greater than zero”). A negative constant in a textbook problem flips the curve’s orientation without changing the enclosed area, so enter its absolute value.
- Read all four lines: the area to four decimal places, the curve type, the polar equation as computed, and the a/b ratio to three decimals. The formula line underneath restates the substitution, π(a² + b²/2) with your numbers in place, so the run can be checked at a glance.
The formula: π(a² + b²/2), and where it comes from
The area inside a limaçon is π(a² + b²/2), for both the cosine and the sine form. It comes from the polar area integral evaluated once, in closed form.
Expand: (a + b·cos θ)² = a² + 2ab·cos θ + b²·cos² θ
Integrate over a full turn: the 2ab·cos θ term integrates to 0, a² gives 2πa², and b²·cos² θ gives πb²
Result: A = ½(2πa² + πb²) = π(a² + b²/2)
That derivation is the standard one for polar area, set out in OpenStax, Calculus Volume 2, section 7.4, Area and Arc Length in Polar Coordinates, where area and arc length are treated as the two basic measurements of a polar curve. Two consequences are worth keeping. First, the formula needs no angle range and no numeric integration: one full turn is baked into it. Second, it was verified against brute-force numeric integration for this page: stepping θ through two million slices and summing ½r² Δθ reproduces π(a² + b²/2) to four decimal places for every worked example in the table below, including the inner-loop case.
The four limacon shapes
The shape is decided entirely by how a compares with b, and the tool classifies it for you on every run. Each row below is a verified run of the calculator above.
| Shape | Condition | a/b ratio | Verified example | Area |
|---|---|---|---|---|
| Cardioid (heart-shaped) | a = b | exactly 1 | r = 1 + 1 cos θ | 4.7124 |
| Dimpled | b < a < 2b | between 1 and 2 | r = 3 + 2 cos θ | 34.5575 |
| Convex | a ≥ 2b | 2 or more | r = 4 + 2 cos θ | 56.5487 |
| Inner loop | a < b | below 1 | r = 1 + 2 cos θ | 9.4248 |
The cardioid row is a special case worth its own double-check: at a = b the formula gives π(a² + a²/2) = 3πa²/2, the textbook cardioid area, and the cardioid area calculator reaches the same number from the cardioid’s own parametrization. At the other end, as b shrinks toward zero against a, the convex row rounds toward a plain circle of radius a; the circle formulas take over in that limit, though this tool requires b above zero so the limaçon character never fully disappears. One display trap: the printed ratio is rounded to three decimals, so a = 1.0001, b = 1 prints a ratio of 1.000 while the classifier, which compares the exact values, correctly calls the curve dimpled, with an area of 4.7130.
Worked examples, checked against this exact tool
Every row is a run of the calculator above, including the sine run that proves the two forms agree. Enter the same values to reproduce any row.
| Run | Equation | Type | Area (sq units) |
|---|---|---|---|
| Worked example | r = 3 + 2 cos θ | Dimpled | 34.5575 |
| Same a and b, sine form | r = 3 + 2 sin θ | Dimpled | 34.5575 |
| Small cardioid | r = 1 + 1 cos θ | Cardioid (heart-shaped) | 4.7124 |
| Larger cardioid | r = 2 + 2 cos θ | Cardioid (heart-shaped) | 18.8496 |
| Boundary convex | r = 4 + 2 cos θ | Convex | 56.5487 |
| Small convex | r = 0.5 + 0.25 cos θ | Convex | 0.8836 |
Doubling both constants quadruples the area, as the two cardioid rows show (4.7124 to 18.8496): area scales with the square of the size, because both a² and b² scale together. The sine row is not a rounding coincidence either; rotating a polar curve cannot change the area it encloses, and the tool’s identical output is the numeric shadow of that fact. For the length of the same curves rather than their area, the arc length calculator covers the companion measurement.
Cosine vs sine form
The two forms enclose exactly the same area for the same a and b. The sine form is the cosine form rotated 90 degrees.
The selector exists because homework problems are stated in one form or the other, and seeing the equation confirmed under your result avoids copying mistakes. Nothing else changes: the classifier, the ratio and the formula all read a and b the same way. If your problem is stated with the variable term first, as r = b·cos θ + a, match by role, not by position in the string: the constant standing alone is a, the coefficient of the trig term is b.
The inner-loop case, stated precisely
When a < b, the printed area is the full-turn polar integral, which counts the inner loop’s area on top of the region inside the outer loop. For r = 1 + 2 cos θ the tool prints 9.4248, and numeric integration splits that figure into 8.8813 swept while the outer loop is traced plus the inner loop’s own 0.5435.
This matters because textbook questions about looped limaçons often ask for something narrower: the area inside the outer loop only (8.8813 in the example), the area of the inner loop alone (0.5435), or the area between the loops (8.8813 minus 0.5435 = 8.3378). None of those is the single number this tool prints. Each needs a piecewise integral over the angles where r is negative, the same set-up used for the area between two functions in Cartesian coordinates, where the region has to be described before it can be integrated. Use this page for the whole-curve area and the classification; set up the piecewise integral when the question names one loop or the gap between them.
What competing calculators miss
Very few dedicated limaçon tools exist, and the ones that do leave the classification, the one thing a student actually needs checked, undone or vague:
- BraveCalculator’s limacon area calculator: the broadest rival, asking for a, b, a start and end angle in degrees, a scale factor and even a material cost per unit area. The extra inputs buy little for a full-turn area, and its shape rules are loose: dimpled “when a > b” with no 2b boundary, and convex “when a ≫ b”, so the dimpled-to-convex line, the exact place students slip, is never drawn.
- Savvy Calculator’s limacon area calculator: uses the same closed form, π(a² + b²/2), and shows the derivation, which matches the consensus. Its tool output is the area alone: no curve type, no a/b ratio, no printed equation to check against the problem statement.
- TrueGeometry’s limacon tutorial: states the area as the integral of (a + b·cos θ) itself over 0 to 2π, dropping the ½r² structure of polar area entirely. That expression has the wrong units and the wrong value; it is a reminder to check any formula against the integral it claims to come from.
- Vedantu’s worked solution: solves “the area inside a limacon” with the parameters in swapped roles (r = b + a·cos θ) and an explicit assumption that the curve does not cross itself, which quietly excludes the inner-loop family that this page handles and splits out in numbers.
- GauthMath’s inner-loop answer: for the area between the loops of r = 1 + 2 cos θ it prints 1.08703288447, which is the antiderivative difference before the one-half factor is applied; the inner loop’s area is 0.5435, verified by direct numeric integration for this page.
- The gap this page fills: the consensus formula with nothing extra to enter, the shape classified by the exact 1 and 2 ratio boundaries, the precise equation printed back, and an honest account of what the number contains when the curve loops.
Frequently asked questions
A limaçon (pronounced “lee-mah-sohn”) is a polar curve with the equation r = a + b·cos θ or r = a + b·sin θ, where a and b are positive constants. It can look like a heart, a dented oval, a smooth oval, or a looped figure.
The tool on this page computes the area as π(a² + b²/2) from the parameters you enter: integrate ½(a + b·cos θ)² over a full turn and the cross term vanishes, leaving πa² + πb²/2.
A cardioid is the heart-shaped limaçon that appears when a = b, for example r = 1 + cos θ. Its area is 3πa²/2, which is what the general formula gives at a = b; the calculator returns 4.7124 square units for a = b = 1.
When a < b the curve traces a small loop inside the larger outer curve, for example r = 2 + 5·cos θ. The loop appears because r goes negative over a range of angles, and those points plot on the opposite side of the origin.
No. The enclosed area is identical; only the orientation changes. Use the selector that matches how your problem is stated. For a = 3 and b = 2 both forms return 34.5575 square units in this tool.
It is the quick classifier: a ratio of exactly 1 means a cardioid, above 1 but below 2 means dimpled, 2 or more means convex, and below 1 means an inner loop. The printed ratio is rounded to three decimals, so a curve very close to a cardioid can show 1.000 while its exact ratio is just above 1 and its type is dimpled.
Not in this tool: both parameters must be greater than zero, and the calculator shows an error otherwise. In the mathematics, a negative constant flips the curve’s orientation but does not change the enclosed area, so enter the absolute value to get the area for a problem stated with a negative constant.
From the polar area integral, one half the integral of r(θ)² from 0 to 2π. Expanding (a + b·cos θ)² gives a² + 2ab·cos θ + b²·cos² θ; the middle term integrates to zero over a full turn, the first gives 2πa² and the last gives πb², so half the total is π(a² + b²/2).
The printed number is the full-turn integral, so it counts the inner loop’s area on top of the region inside the outer loop. For r = 1 + 2·cos θ the tool prints 9.4248: that is 8.8813 swept while tracing the outer loop plus the inner loop’s 0.5435. If a problem asks for one loop only, or the area between the loops, it needs a piecewise integral instead.
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References
- OpenStax. Calculus Volume 2, Section 7.4: Area and Arc Length in Polar Coordinates. Accessed October 22, 2026.
- OpenStax. Calculus Volume 2, Section 7.3: Polar Coordinates. Accessed October 22, 2026.
- Dawkins, Paul. Calculus II: Area with Polar Coordinates, Paul’s Online Math Notes, Lamar University. Accessed October 22, 2026.
- MacTutor History of Mathematics, University of St Andrews. Limaçon of Pascal. Accessed October 22, 2026.
- MacTutor History of Mathematics, University of St Andrews. Cardioid. Accessed October 22, 2026.