The cardioid: equation, graph and properties

A cardioid is the heart-shaped curve traced by a point on a circle rolling around another circle of the same size. In polar coordinates it is simply r = a(1 + cos θ), and it turns up in coffee cups, microphones and the Mandelbrot set.

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The cardioid r = 1 + cos θ. It is traced by a point on a circle of radius ½ rolling around the fixed circle of radius ½ shown dashed.

Equations to type

Polar

r = 1 + cos(theta)

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Parametric

((1 + cos(t)) cos(t), (1 + cos(t)) sin(t))

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Cartesian

(x^2 + y^2 - x)^2 = x^2 + y^2

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At a glance

Polar equation: r = a(1 + cos θ)
Area: 3πa²/2
Length: 8a
Cusp: At the origin

A circle rolling around a circle

Fix a circle of radius a/2 and roll another circle of the same radius around the outside of it without slipping. A point on the rolling circle traces a cardioid. It touches the fixed circle once per trip, which makes the single cusp. The cardioid is the epicycloid with one cusp; rolling smaller circles gives epicycloids with more cusps.

Area and length

Area in polar coordinates is ½ ∫ r² dθ. For r = 1 + cos θ this is ½ ∫ (1 + cos θ)² dθ from 0 to 2π = 3π/2, about 4.712. In general a cardioid of size a encloses 3πa²/2, exactly six times the area of the rolling circle.

Its total length is 8a: for r = 1 + cos θ, exactly 8. The arc length formula ∫ √(r² + (dr/dθ)²) dθ simplifies to ∫ 2|cos(θ/2)| dθ, which integrates neatly.

Where cardioids appear

The bright caustic of light reflected inside a cup or ring is a cardioid (or the closely related nephroid).
A cardioid microphone picks up sound most strongly from the front, with a sensitivity pattern shaped like r = 1 + cos θ.
The main body of the Mandelbrot set is exactly a cardioid.