Theorem: Length of Perimeter of Cardioid

Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
:$r = 2 a \paren {1 + \cos \theta}$
where $a > 0$.
The length of the perimeter of $C$ is $16 a$.

Proof:
Let $\LL$ 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:
:$\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
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 for Derivatives
}}
{{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 = \LL
      | 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}}