Limaçon — Definition, Formula & Graph

Q: What is the difference between a limaçon and a cardioid?

A cardioid is a special case of the limaçon where b = a, producing the heart-shaped curve r = a(1 + cos θ). When b ≠ a, the limaçon takes on other shapes: an inner loop (b < a), a dimple (a < b < 2a), or a convex oval (b ≥ 2a). Every cardioid is a limaçon, but not every limaçon is a cardioid.

Q: How do you determine the shape of a limaçon from its equation?

Compare the ratio b/a. If b/a < 1, the curve has an inner loop. If b/a = 1, it is a cardioid. If 1 < b/a < 2, the curve is dimpled (indented but no loop). If b/a ≥ 2, the curve is convex with no dimple. These four cases cover all limaçon shapes.

Q: What does 'limaçon' mean and how do you pronounce it?

Limaçon comes from the French word for 'snail,' reflecting the curve's spiral-like appearance. It is pronounced 'LEE-mah-sohn' (with a soft 's' sound for the ç). The curve was first studied by Étienne Pascal (father of Blaise Pascal) in the 17th century.

Limaçon

A famliy of related curves usually expressed in polar coordinates. The cardioid is a special kind of limaçon.

Limaçon: r = b + a cos θ (horizontal, pictured below) or r = b + a sin θ (vertical)

Three limaçon curves: looped (b<a), dimpled (a<b<2a), and convex (2a≤b), shown in polar coordinates.

Note: If a = b the curve is a cardioid.

See also

Convex, polar curves

Key Formula

r = b + a\cos\theta \quad \text{or} \quad r = b + a\sin\theta

Where:

$r$ = The radial distance from the origin to a point on the curve
$\theta$ = The angle measured from the positive x-axis (in radians or degrees)
$a$ = Controls the size of the loop or dimple; relates to the distance from the origin to the focus
$b$ = Controls the overall offset of the curve from the origin

Worked Example

Problem: Sketch and classify the limaçon r = 1 + 2 cos θ. Find the values of r at key angles and determine whether the curve has an inner loop.

Step 1: Identify a and b. Here b = 1 and a = 2. Since b/a = 1/2 < 1, the limaçon has an inner loop.

b = 1,\quad a = 2,\quad \frac{b}{a} = \frac{1}{2} < 1

Step 2: Evaluate r at key angles. At θ = 0: r = 1 + 2(1) = 3. At θ = π/2: r = 1 + 2(0) = 1. At θ = π: r = 1 + 2(−1) = −1.

r(0) = 3,\quad r\!\left(\tfrac{\pi}{2}\right) = 1,\quad r(\pi) = -1

Step 3: The negative value r = −1 at θ = π confirms an inner loop. When r becomes negative, the point is plotted in the opposite direction, creating a small loop inside the larger curve.

Step 4: Find where the inner loop begins and ends by setting r = 0.

0 = 1 + 2\cos\theta \implies \cos\theta = -\frac{1}{2} \implies \theta = \frac{2\pi}{3},\; \frac{4\pi}{3}

Step 5: The inner loop exists for θ between 2π/3 and 4π/3. The curve extends farthest from the origin (r = 3) at θ = 0 and the deepest point of the inner loop (r = −1) is at θ = π.

\text{Inner loop: } \frac{2\pi}{3} \leq \theta \leq \frac{4\pi}{3}

Answer: The curve r = 1 + 2 cos θ is a limaçon with an inner loop. It reaches a maximum distance of 3 from the origin at θ = 0, and the inner loop extends from θ = 2π/3 to θ = 4π/3.

Another Example

This example applies calculus (polar area integration) to the same limaçon, showing how to compute the area of the inner loop — a common exam problem in AP Calculus BC and college calculus courses.

Problem: Find the area enclosed by the inner loop of the limaçon r = 1 + 2 cos θ.

Step 1: From the first example, the inner loop occurs where r ≤ 0, between θ = 2π/3 and θ = 4π/3.

\cos\theta = -\frac{1}{2} \implies \theta = \frac{2\pi}{3},\; \frac{4\pi}{3}

Step 2: Use the polar area formula over the inner loop interval.

A = \frac{1}{2}\int_{2\pi/3}^{4\pi/3} r^2\, d\theta = \frac{1}{2}\int_{2\pi/3}^{4\pi/3} (1 + 2\cos\theta)^2\, d\theta

Step 3: Expand the integrand: (1 + 2 cos θ)² = 1 + 4 cos θ + 4 cos²θ. Replace cos²θ using the double-angle identity.

1 + 4\cos\theta + 4\cos^2\theta = 1 + 4\cos\theta + 2(1 + \cos 2\theta) = 3 + 4\cos\theta + 2\cos 2\theta

Step 4: By symmetry about the polar axis, double the integral from 2π/3 to π. Integrate term by term.

A = 2 \cdot \frac{1}{2}\int_{2\pi/3}^{\pi}\left(3 + 4\cos\theta + 2\cos 2\theta\right)d\theta = \int_{2\pi/3}^{\pi}\left(3 + 4\cos\theta + 2\cos 2\theta\right)d\theta

Step 5: Evaluate: [3θ + 4 sin θ + sin 2θ] from 2π/3 to π. At θ = π: 3π + 0 + 0 = 3π. At θ = 2π/3: 2π + 4(√3/2) + sin(4π/3) = 2π + 2√3 − √3/2 = 2π + 3√3/2.

A = 3\pi - \left(2\pi + \frac{3\sqrt{3}}{2}\right) = \pi - \frac{3\sqrt{3}}{2}

Answer: The area of the inner loop is π − (3√3)/2 ≈ 0.544 square units.

Frequently Asked Questions

What is the difference between a limaçon and a cardioid?

A cardioid is a special case of the limaçon where b = a, producing the heart-shaped curve r = a(1 + cos θ). When b ≠ a, the limaçon takes on other shapes: an inner loop (b < a), a dimple (a < b < 2a), or a convex oval (b ≥ 2a). Every cardioid is a limaçon, but not every limaçon is a cardioid.

How do you determine the shape of a limaçon from its equation?

Compare the ratio b/a. If b/a < 1, the curve has an inner loop. If b/a = 1, it is a cardioid. If 1 < b/a < 2, the curve is dimpled (indented but no loop). If b/a ≥ 2, the curve is convex with no dimple. These four cases cover all limaçon shapes.

What does 'limaçon' mean and how do you pronounce it?

Limaçon comes from the French word for 'snail,' reflecting the curve's spiral-like appearance. It is pronounced 'LEE-mah-sohn' (with a soft 's' sound for the ç). The curve was first studied by Étienne Pascal (father of Blaise Pascal) in the 17th century.

Limaçon vs. Cardioid

	Limaçon	Cardioid
Definition	A family of polar curves r = b + a cos θ for any positive a and b	A specific limaçon where b = a, giving a heart shape
Formula	r = b + a cos θ (general)	r = a(1 + cos θ) or r = a + a cos θ
Inner loop?	Yes, when b < a	Never — passes through the origin instead
Shape variations	Looped, cardioid, dimpled, or convex	Always heart-shaped with one cusp
Enclosed area	Depends on a and b; requires integration for inner loop cases	A = (3/2)πa²

Why It Matters

Limaçons are one of the most common families of polar curves studied in precalculus and calculus courses. They appear frequently on AP Calculus BC exams, where students must sketch curves, find enclosed areas, and determine intersections. Understanding how the ratio b/a controls the curve's shape builds strong intuition for polar graphing and parametric equations.

Common Mistakes

Mistake: Confusing which parameter controls the loop. Students sometimes think increasing b creates a loop.

Correction: A loop appears when b < a (i.e., b/a < 1). Increasing b relative to a actually removes the loop and makes the curve more convex. Remember: the loop exists because r becomes negative, which happens when b is too small to offset the negative values of a cos θ.

Mistake: Forgetting that cos θ produces a horizontal limaçon while sin θ produces a vertical one.

Correction: The equation r = b + a cos θ is symmetric about the polar axis (horizontal), while r = b + a sin θ is symmetric about the line θ = π/2 (vertical). Choose the correct form based on the curve's orientation.

Related Terms

Polar Coordinates — The coordinate system used to define limaçons
Cardioid — Special case of a limaçon where a = b
Polar Curves — The broader category that includes limaçons
Curve — General term for continuous paths in the plane
Convex — Describes the limaçon shape when b ≥ 2a
Rose Curve — Another common family of polar curves