A cardioid (from the Greek καρδία "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. The name was coined by de Castillon in 1741, but had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. The arc length of the curve would be $\displaystyle s=8a\sin^2\left ( \frac{1}{4}t \right )$. The perimeter would be L=8a and the area would be $\displaystyle A=\frac{3}{2}\pi a^2$. Click "Read More" to see the interactive graphic. | Equation: Cartesian Equation: $\displaystyle \left ( x^2 +y^2-2ax\right )^2=4a^2\left ( x^2+y^2 \right )$ Parametric Equations: $\displaystyle x=a\cos{t}\left ( 1-\cos{t} \right )$ $\displaystyle y=a\sin{t}\left ( 1-\cos{t} \right )$ Polar Equation: $\displaystyle r=a\left ( 1-\cos{\theta} \right )$ OR $\displaystyle r=2b\left ( 1-\cos{\theta} \right )$ Graphs: |

**Interactive Graphic**

**References:**

*Cardioid*. N.p., Jan. 1997. Web. 09 Nov. 2013.

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*Wikipedia*. Wikimedia Foundation, 19 Oct. 2013. Web. 09 Nov. 2013.

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*Wolfram|Alpha*. Wolfram, 9 Nov. 2013. Web. 09 Nov. 2013.

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*MathWorld*. Wolfram, 5 July 2008. Web. 9 Nov. 2013.