Cardioid

A cardioid is an epicycloid with one cusp; a standard polar equation is $r=a(1+\cos \theta )$.

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.

For $a>0$ and $0<c<2a$, consider the region $c\le r\le a(1+\cos \varphi )$ in polar coordinates. It exists precisely for $|\varphi |\le \beta$, where $\beta =\arccos (c/a-1)\in (0,\pi )$. The polar area element $rdrd\varphi$ givesExpanding the square and integrating its even terms gives the title formula. This cap lies inside the cardioid and outside the concentric circle, so reversing the radial bounds would select the wrong region or a signed area.