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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.

Differentiate $x=r\cos \theta$, $y=r\sin \theta$:Squaring and adding cancels the mixed terms and uses ${\cos }^2\theta +{\sin }^2\theta =1$, givingFor this cardioid, the polar arc length integrand isThe cosine inside the absolute value changes sign at $\theta =\pi$. Therefore the perimeter isin coordinate length units. The isolated cusp does not obstruct this piecewise smooth arc length integral.