Length of Perimeter of Cardioid

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Theorem

Consider the cardioid $C$ embedded in a polar plane given by its polar equation:

$r = 2 a \paren {1 + \cos \theta}$


The length of the perimeter of $C$ is $16 a$.


Proof

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:

\(\displaystyle \frac {\d x} {\d \theta}\) \(=\) \(\displaystyle 2 a \map {\frac \d {\d \theta} } {\cos \theta + \cos^2 \theta}\) rearranging
\(\displaystyle \) \(=\) \(\displaystyle -2 a \paren {\sin \theta + 2 \cos \theta \sin \theta}\) Derivative of Cosine Function, Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle -2 a \paren {\sin \theta + \sin 2 \theta}\) Double Angle Formula for Sine
\(\displaystyle \frac {\d y} {\d \theta}\) \(=\) \(\displaystyle 2 a \map {\frac \d {\d \theta} } {\sin \theta + \sin \theta \cos \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle 2 a \paren {\cos \theta + \cos^2 \theta - \sin^2 \theta}\) Derivative of Sine Function, Product Rule
\(\displaystyle \) \(=\) \(\displaystyle 2 a \paren {\cos \theta + \cos 2 \theta}\) Double Angle Formula for Cosine


Thus:

\(\displaystyle \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2}\) \(=\) \(\displaystyle \sqrt {4 a^2 \paren {\paren {\sin \theta + \sin 2 \theta}^2 + \paren {\cos \theta + \cos 2 \theta}^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sqrt {\sin^2 \theta + 2 \sin \theta \sin 2 \theta + \sin^2 2 \theta + \cos^2 \theta + 2 \cos \theta \cos 2 \theta + \cos^2 2 \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sqrt {2 + 2 \sin \theta \sin 2 \theta + 2 \cos \theta \cos 2 \theta}\) Sum of Squares of Sine and Cosine in $2$ instances
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sqrt {2 + 2 \sin \theta \paren {2 \sin \theta \cos \theta} + 2 \cos \theta \paren {\cos^2 \theta - \sin^2 \theta} }\) Double Angle Formulas
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sqrt {2 + 4 \sin^2 \theta \cos \theta + 2 \cos^3 \theta - 2 \sin^2 \theta \cos \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sqrt {2 + 2 \sin^2 \theta \cos \theta + 2 \cos^3 \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sqrt {2 + 2 \cos \theta \paren {\sin^2 \theta + \cos^2 \theta} }\)
\(\displaystyle \) \(=\) \(\displaystyle 4 a \sqrt {\dfrac {1 + \cos \theta} 2}\) Sum of Squares of Sine and Cosine and extracting factor
\((1):\quad\) \(\displaystyle \) \(=\) \(\displaystyle 4 a \cos \dfrac \theta 2\) Half Angle Formula for Cosine


\(\displaystyle \mathcal L\) \(=\) \(\displaystyle \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta\) Area between Radii and Curve in Polar Coordinates
\(\displaystyle \) \(=\) \(\displaystyle \int_{-\pi}^\pi 4 a \cos \dfrac \theta 2 \rd \theta\) from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle 4 a \intlimits {2 \sin \dfrac \theta 2} {-\pi} \pi\) Primitive of $\cos a x$
\(\displaystyle \) \(=\) \(\displaystyle 8 a \paren {\sin \dfrac \pi 2 - \sin \dfrac {-\pi} 2}\) evaluation between $-\pi$ and $\pi$
\(\displaystyle \) \(=\) \(\displaystyle 8 a \paren {1 - \paren {-1} }\) Sine of Right Angle, Sine Function is Odd
\(\displaystyle \) \(=\) \(\displaystyle 16 a\)

$\blacksquare$


Sources