Cardioid
College/University
Definition
Describes a curve that is heart-like in shape. It is created when a circle rolls around a fixed circle that has the same radius. A cardioid has 3 parallel tangents and a cusp (the intersection of two branches of a curve). Named after the Latin word for heart, cordis.
Worked examples
\(r = a(1 + \cos\theta)\)
Polar equation of a cardioid with cusp at the origin and symmetry about the polar axis.
\(r = 2 + 2\sin\theta\)
A cardioid shifted so the cusp points downward; notice the equal coefficients create the heart shape.
Common mistakes
- \(r = 1 + 2\cos\theta\) is a cardioid → \(r = a(1 + \cos\theta)\) requires equal coefficients Unequal coefficients produce a limaçon, not a cardioid.
- A cardioid has no cusp because it's smooth everywhere → A cardioid has exactly one cusp where the curve meets itself The cusp is a defining feature where the two branches intersect at a sharp point.
- The three parallel tangents can be at any angle → The three parallel tangents are perpendicular to the axis of symmetry Their orientation is fixed by the cardioid's symmetry axis.
Where you'll use it next
Cardioids appear in precalculus polar graphing, calculus area and arc-length problems, and acoustics (microphone pickup patterns). You'll see them again studying parametric equations and cycloid curves.
Found in 1 StudyPug lesson
Converting Polar to Cartesian Equations: A Comprehensive Guide
UniversityUniversityCalculus 2
Unlock the power of coordinate conversion! Learn to effortlessly transform polar equations into Cartesian form. Master this crucial math skill and enhance your problem-solving abilities.
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026