Area and perimeter of a cardioid ($\mathcal{A}=3\pi a^2/2,\ell =8a$)

For $a>0$, the cardioid $r=a(1+\cos \theta )$ traces its boundary once over $0\le \theta \le 2\pi$. The polar area element givesThe polar arc length integrand is $2a|\cos (\theta /2)|$. Splitting at the cusp parameter $\theta =\pi$ gives perimeter $8a$. The powers of $a$ distinguish the scaling of area and length.