এখানে যে যে সূত্র ব্যাবহার হয়েছে:

1. To find the value of $\theta$ from the non-convertible implicit function: \[ \text{Segment Area:} \quad A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta) \] We used the Newton-Raphson method. The iterative formula for Newton's method is: \[ \theta_{n+1} = \theta_n - \frac{f(\theta_n)}{f'(\theta_n)} \] where $f(\theta)$ is the given function, and $f'(\theta)$ is its derivative. For $f(\theta) = A_{\text{segment}} - \frac{1}{2}r^2(\theta - \sin\theta)$, the derivative is: \[ f'(\theta) = \frac{d}{d\theta} \left( A_{\text{segment}} - \frac{1}{2}r^2(\theta - \sin\theta) \right) \] To find the derivative, apply the chain rule: \[ f'(\theta) = \frac{d}{d\theta} A_{\text{segment}} - \frac{d}{d\theta} \frac{1}{2}r^2(\theta - \sin\theta) \] After differentiation and simplification, we get: \[ f'(\theta) = r^2\left(1 - \cos\theta\right) \] Now, we can iterate using the Newton-Raphson formula until convergence: \[ \theta_{n+1} = \theta_n - \frac{A_{\text{segment}} - \frac{1}{2}r^2(\theta_n - \sin\theta_n)}{r^2\left(1 - \cos\theta_n\right)} \]

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2. Area of a circle: \[ \text{Area} = \pi \times R^2 \] Where: \[ R = \text{radius} \] \[ D = \text{diameter} = 2 \times R \] \[ P = \text{perimeter} = 2 \times \pi \times R \]
3. \[ \text{Radius, r} = \frac{c^2}{8h_{\text{segment}}} + \frac{h_{\text{segment}}}{2} \] \[ \text{Segment Area:} \quad A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta) \] \[ \text{Segment Height:} \quad h_{\text{segment}} = r - \sqrt{r^2 - \left(\frac{c}{2}\right)^2} \] \[ \text{Chord Length:} \quad c = 2r\sin\left(\frac{\theta}{2}\right) \] \[ \text{Arc Length:} \quad s = r\theta \] \[ \text{Sector Area:} \quad A_{\text{sector}} = \frac{1}{2} r^2 \theta \]