# Length of Perimeter of Cardioid

## 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 $\mathcal L$ denote the length of the perimeter of $C$.

The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.

$\displaystyle \mathcal L = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$

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 for Derivatives $\displaystyle$ $=$ $\displaystyle -2 a \paren {\sin \theta + \sin 2 \theta}$ 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 $\text {(1)}: \quad$ $\displaystyle$ $=$ $\displaystyle 4 a \cos \dfrac \theta 2$ 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 $\displaystyle$ $=$ $\displaystyle 16 a$

$\blacksquare$

## Proof 2

Let $\mathcal L$ denote the length of the perimeter of $C$.

The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.

$\displaystyle \mathcal L = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$

where:

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

Note that we have:

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

We therefore have:

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

$\blacksquare$