Length of Perimeter of Cardioid

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Theorem

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

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

where $a > 0$.


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


Proof 1

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:

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


Thus:

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


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

$\blacksquare$


Proof 2

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 Polar Curve:

$\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$

where:

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


Note that we have:

\(\ds \frac {\d r} {\d \theta}\) \(=\) \(\ds 2 a \frac \d {\d \theta} \paren {1 + \cos \theta}\)
\(\ds \) \(=\) \(\ds -2 a \sin \theta\) Sum Rule for Derivatives, Derivative of Cosine Function

We therefore have:

\(\ds \LL\) \(=\) \(\ds \int_{-\pi}^\pi \sqrt {4 a^2 \paren {1 + \cos \theta}^2 + 4 a^2 \sin^2 \theta} \rd \theta\)
\(\ds \) \(=\) \(\ds 2 a \int_{-\pi}^\pi \sqrt {1 + 2 \cos \theta + \cos^2 \theta + \sin^2 \theta} \rd \theta\) extracting a factor of $\sqrt {4 a^2} = 2 a$, noting that $a \ge 0$
\(\ds \) \(=\) \(\ds 2 a \int_{-\pi}^\pi \sqrt {2 + 2 \cos \theta} \rd \theta\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 4 a \int_0^\pi \sqrt {4 \cos^2 \frac \theta 2} \rd \theta\) Definite Integral of Even Function, Double Angle Formula for Cosine: Corollary 1
\(\ds \) \(=\) \(\ds 8 a \int_0^\pi \size {\cos \frac \theta 2} \rd \theta\)
\(\ds \) \(=\) \(\ds 8 a \int_0^\pi \cos \frac \theta 2 \rd \theta\) Definition of Absolute Value, noting that $\cos \theta \ge 0$ for $0 \le \theta \le \dfrac \pi 2$
\(\ds \) \(=\) \(\ds 16 a \intlimits {\sin \frac \theta 2} 0 \pi\) Primitive of $\cos a x$
\(\ds \) \(=\) \(\ds 16 a \paren {\sin \frac \pi 2 - \sin 0}\)
\(\ds \) \(=\) \(\ds 16 a\) Sine of Right Angle, Sine of Zero is Zero

$\blacksquare$


Sources

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