Length of Perimeter of Cardioid

Theorem
Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
 * $r = 2 a \paren {1 + \cos \theta}$

The length of the perimeter of $C$ is $16 a$.

Proof
Let $\mathcal L$ denote the length of the perimeter of $C$.

The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.

From Arc Length for Parametric Equations:


 * $\displaystyle \mathcal L = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$

where, from Equation of Cardioid:
 * $\begin {cases}

x & = 2 a \cos \theta \paren {1 + \cos \theta} \\ y & = 2 a \sin \theta \paren {1 + \cos \theta} \end {cases}$

We have:

Thus: