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A plane algebraic curve of order four which is described by a point of a circle of radius rolling on a circle with the same radius ; an epicycloid with modulus . The equation of the cardioid in polar coordinates is:

In Cartesian coordinates it is:

The arc length from the cusp is:

The radius of curvature is:

The area bounded by the curve equals . The length of the curve is . The cardioid is a conchoid of the circle, a special case of a Pascal limaçon and a sinusoidal spiral.

Figure: c020390a


[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)



[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Cardioid. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article