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Field: Geometry
Image Created By: The Math Book


A cardioid is a curve which resembles a heart. Its name is derived from Greek where kardia means heart and eidos means shape, though it is actually shaped more like the outline of the cross section of an apple.

Basic Description

In geometry, a cardioid is the curve traced by a point on the circumference of a circle that rolls around the circumference of another equal circle.

Cardioid 1.gif

A More Mathematical Explanation

The curve is given by:

  • Cartesian equation ({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2}), where '"`UNIQ--math-00 [...]

The curve is given by:

  • Cartesian equation ({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2}), where a is the radius of the moving circle.
  • Polar equation r = a (1 - {\cos} {\theta})
  • Parametric equation

x = a  {\cos}  t (1 - {\cos t})

y = a  {\sin}  t (1 - {\cos t})


  • It has a cusp at the origin.
  • There are exactly three tangents to the cardioid with any given gradient
  • The tangents at the ends of any chord through the cusp point are at right angles
  • The length of any chord through the cusp point is 2a

Generating a Cardioid

Draw a circle C, and pick a fixed point A on it. Then, draw a set of circles centered on the circumference of C and passing through A. The envelop of the chords of these circles is a cardioid, as in the main image. If the fixed point A is not on the circle, then the figure becomes a limacon

The Cardioid in Real Life

An instance where one could see a cardioid is when looking into a cup of coffee. The caustic seen at the bottom of a cup of coffee could be a cardioid, depending on the angle of light relative to the bottom of the cup.

Also, all unidirectional microphones are cardioid-shaped.

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