Cardioid
The heart-shaped curve r = a(1 + cos θ) traced by a point on a circle rolling around an equal circle.
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Formulas and key values
- Polar equation
- Cartesian equation
- PerimeterIn the graph = 8
- AreaIn the graph ≈ 4.712
- Cusps
a sets the size: the cusp is at the origin and the far end at (2a, 0).
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Start the video01 · Definition
What is the Cardioid?
The cardioid is the path of a point on a circle rolling around a second, equal circle — the simplest epicycloid. Its polar equation r = a(1 + cos θ) makes the heart shape explicit: the radius doubles at θ = 0 and collapses to a cusp at θ = π. The name, from the Greek for “heart”, was given by Giovanni Salvemini de Castillon in 1741.
The cardioid keeps appearing where circles interact: it is the envelope of light rays reflected inside a coffee cup (a caustic), the boundary of the main bulb of the Mandelbrot set, and the pickup pattern of the microphones named after it.
02 · Properties
Key properties
- Arc length: 8a. Enclosed area: (3/2)πa².
- It is the epicycloid with one cusp (rolling circle equal to the fixed circle).
- It is also the special limaçon r = b + a·cos θ with b = a.
- The caustic of a circle with a light source on its rim is a cardioid.
- Maximum width 2a occurs along the axis of symmetry.
03 · Questions
Cardioid: common questions
- What is the equation of the Cardioid?
- Polar equation: r = a(1 + cos θ); Cartesian equation: (x^2 + y^2 - ax)^2 = a^2(x^2 + y^2). a sets the size: the cusp is at the origin and the far end at (2a, 0).
- What is the perimeter of the Cardioid?
- Perimeter: L = 8a. a sets the size: the cusp is at the origin and the far end at (2a, 0).
- What is the area of the Cardioid?
- Area: A = (3/2)π a^2. a sets the size: the cusp is at the origin and the far end at (2a, 0).
04 · Wonders
See it in a math wonder
05 · Keep exploring