Cardioid Area Calculator

About this calculator

A cardioid is the heart-shaped curve traced in polar coordinates by r = a(1 + cos θ), where a is a scale parameter — it's also the exact path traced by a point on a circle as that circle rolls around another circle of the same radius (a special case of an epicycloid with a single cusp). Its enclosed area works out to (3/2)πa², and its perimeter (arc length) is exactly 8a.

This calculator computes both quantities directly from a, saving the polar double-integral that would otherwise be needed to derive the area. Math students and instructors covering polar coordinates and parametric curves use it to check textbook problems, and the same r = a(1 + cos θ) family shows up in antenna radiation-pattern diagrams and acoustics, where cardioid-shaped directivity patterns describe microphones that pick up sound mainly from one direction.