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: {{begin-eqn}} {{eqn | l = \frac {\d x} {\d \theta}       | r = 2 a \map {\frac \d {\d \theta} } {\cos \theta + \cos^2 \theta}       | c = rearranging }} {{eqn | r = -2 a \paren {\sin \theta + 2 \cos \theta \sin \theta}       | c = Derivative of Cosine Function, Chain Rule for Derivatives }} {{eqn | r = -2 a \paren {\sin \theta + \sin 2 \theta}       | c = Double Angle Formula for Sine }} {{eqn | l = \frac {\d y} {\d \theta}       | r = 2 a \map {\frac \d {\d \theta} } {\sin \theta + \sin \theta \cos \theta}       | c =  }} {{eqn | r = 2 a \paren {\cos \theta + \cos^2 \theta - \sin^2 \theta}       | c = Derivative of Sine Function, Product Rule }} {{eqn | r = 2 a \paren {\cos \theta + \cos 2 \theta}       | c = Double Angle Formula for Cosine }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2}       | r = \sqrt {4 a^2 \paren {\paren {\sin \theta + \sin 2 \theta}^2 + \paren {\cos \theta + \cos 2 \theta}^2} }       | c =  }} {{eqn | r = 2 a \sqrt {\sin^2 \theta + 2 \sin \theta \sin 2 \theta + \sin^2 2 \theta + \cos^2 \theta + 2 \cos \theta \cos 2 \theta + \cos^2 2 \theta}       | c =  }} {{eqn | r = 2 a \sqrt {2 + 2 \sin \theta \sin 2 \theta + 2 \cos \theta \cos 2 \theta}       | c = Sum of Squares of Sine and Cosine in $2$ instances }} {{eqn | r = 2 a \sqrt {2 + 2 \sin \theta \paren {2 \sin \theta \cos \theta} + 2 \cos \theta \paren {\cos^2 \theta - \sin^2 \theta} }       | c = Double Angle Formulas }} {{eqn | r = 2 a \sqrt {2 + 4 \sin^2 \theta \cos \theta + 2 \cos^3 \theta - 2 \sin^2 \theta \cos \theta}       | c =  }} {{eqn | r = 2 a \sqrt {2 + 2 \sin^2 \theta \cos \theta + 2 \cos^3 \theta}       | c =  }} {{eqn | r = 2 a \sqrt {2 + 2 \cos \theta \paren {\sin^2 \theta + \cos^2 \theta} }       | c =  }} {{eqn | r = 4 a \sqrt {\dfrac {1 + \cos \theta} 2}       | c = Sum of Squares of Sine and Cosine and extracting factor }} {{eqn | n = 1       | r = 4 a \cos \dfrac \theta 2       | c = Half Angle Formula for Cosine }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \mathcal L       | r = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta       | c = Area between Radii and Curve in Polar Coordinates }} {{eqn | r = \int_{-\pi}^\pi 4 a \cos \dfrac \theta 2 \rd \theta       | c = from $(1)$ }} {{eqn | r = 4 a \intlimits {2 \sin \dfrac \theta 2} {-\pi} \pi       | c = Primitive of $\cos a x$ }} {{eqn | r = 8 a \paren {\sin \dfrac \pi 2 - \sin \dfrac {-\pi} 2}       | c = evaluation between $-\pi$ and $\pi$ }} {{eqn | r = 8 a \paren {1 - \paren {-1} }       | c = Sine of Right Angle, Sine Function is Odd }} {{eqn | r = 16 a       | c =  }} {{end-eqn}} {{qed}}