Cardioid Area Calculator

How it is calculated

The curve is r = a(1 + cos θ), where a > 0 is the single scale parameter. Polar area is ½ ∫ r² dθ over a full turn (0 to 2π). Expanding r² = a²(1 + 2cos θ + cos²θ) and integrating gives Area = 3πa²/2 (1.5 × π × a²). The tool also reports the arc length, which for a cardioid is exactly Perimeter = 8a. Both results are shown to four decimals. Area is in the square of whatever unit you use for a, and perimeter is in the same unit as a. A zero or negative a returns no result. Note that a is not the width of the heart: the curve reaches 2a at its farthest point from the cusp.

Common mistakes

• Treating a as the radius of the whole heart. The longest distance from the cusp is 2a, not a.
• Applying the circle formula πr² with r = 2a. That gives 4πa², far more than the true 1.5πa².
• Mixing units: if a is in cm, the area is in cm² and the perimeter in cm; do not read both as the same unit.
• Using a different cardioid form such as r = 2a(1 + cos θ) without adjusting. In that form the scale parameter is 2a, so enter the full coefficient that multiplies (1 + cos θ).
• Expecting the perimeter to scale like the area. It scales linearly with a, the area with a².