Length of Perimeter of Cardioid/Proof 2

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 Polar Curve:


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

where:


 * $r = 2a \paren {1 + \cos \theta}$

Note that we have:

We therefore have: