Estimating the Proportion of a Quarter Circle in a Square Using Monte Carlo Method

The Monte Carlo method is employed to estimate the proportion of a quarter circle within a square, which in turn allows for calculating the value of π. As a numerical computation technique based on random number statistics, this method involves generating a large number of random points within a square region. The key algorithmic steps include: counting points falling inside the quarter circle (denoted as A) and the total number of generated points (denoted as B). The proportion of the quarter circle area to the square area is approximated by A/B. Since the quarter circle's area is πr²/4 and the square's area is r², the area ratio theoretically equals π/4. Thus, π can be approximated as 4 times the ratio A/B. In code implementation, this typically involves: 1. Generating uniform random coordinates (x, y) within the range [0, r] 2. Applying the circle equation condition x² + y² ≤ r² to identify points inside the quarter circle 3. Accumulating counts for valid points and total points To validate the method's accuracy, simulations with 10,000 and 50,000 random points are performed. With fewer sampling points, results may show noticeable deviation due to statistical variance. However, as the number of iterations increases, the approximation converges toward π's true value. The implementation can be enhanced through iterative sampling with progressively larger point counts to improve precision, leveraging the law of large numbers for statistical stability.