The formula to calculate the Cardioid Area is:
\[ CRA = 6 \cdot \pi \cdot a^2 \]
Where:
Let's say the value of \( a \) is 2. The Cardioid Area would be calculated as follows:
\[ CRA = 6 \cdot \pi \cdot 2^2 = 6 \cdot \pi \cdot 4 = 24 \cdot \pi \approx 75.40 \]
So, the Cardioid Area is approximately 75.40 square units.
The Cardioid Area (CRA) is the area enclosed by a cardioid, which is a heart-shaped curve described by a specific polar equation. It is calculated by multiplying the square of the value \( a \) by 6 times pi.
Formula: \( A = \frac{1}{2} \int_{0}^{2\pi} r^2 , d\theta \)
Example: \( A = \frac{1}{2} \int_{0}^{2\pi} (1 + \cos\theta)^2 , d\theta \)
Formula: \( L = \int_{0}^{2\pi} \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2} , d\theta \)
Example: \( L = \int_{0}^{2\pi} \sqrt{\left(\frac{d(1 + \cos\theta)}{d\theta}\right)^2 + (1 + \cos\theta)^2} , d\theta \)
Formula: \( A = \pi R^2 - \frac{1}{2} \int_{0}^{2\pi} r^2 , d\theta \)
Example: \( A = \pi (2)^2 - \frac{1}{2} \int_{0}^{2\pi} (1 + \cos\theta)^2 , d\theta \)
