Word Count : 479

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To find the perimeter of the cardioid given by the polar equation \( r = a(1 - \cos \theta) \), we can use the formula for the arc length in polar coordinates:

\[ L = \int_0^{2\pi} \sqrt{\left( \frac{dr}{d\theta} \right)^2 + r^2} \, d\theta \]

First, we need to find \(\frac{dr}{d\theta}\):

\[ r = a(1 - \cos \theta) \]

\[ \frac{dr}{d\theta} = a \sin \theta \]

Now, substitute \( r \) and \( \frac{dr}{d\theta} \) into the arc length formula:

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