Enter how many points to use:
Estimated Pi: 0.0000000000000000000000000
Actual Pi (to 25 places): 3.1415926535897931159979635
Error Margin: 0.000000
Accuracy: 0.000000%
How It Works: Estimating Pi with Monte Carlo Simulation
Welcome to the Monte Carlo simulation. It is designed to estimate the value of Pi by doing the following:
- The Setup: A large circle is perfectly inscribed within a square. For example, the square on this page is 500x500 pixels, so the circle has a radius of 250 pixels.
- Random Point Generation: The simulation randomly generates a specified number of points within the square. Each point has an equal chance of landing anywhere inside the square.
- Point Classification: Each point is checked to see if it lies inside the circle or outside it:
- Inside the Circle: If the distance from the point to the center of the circle is less than or equal to the radius.
- Outside the Circle: If the distance is greater than the radius.
- Pi Estimation: The ratio of points inside the circle to the total number of points approximates the area ratio of the circle to the square. Since the area of the circle is \( \pi R^2 \) and the area of the square is \( (2R)^2 = 4R^2 \), the ratio is \( \frac{\pi}{4} \). Therefore, Pi is estimated as:
\[ \pi \approx 4 \times \left( \frac{\text{Points Inside Circle}}{\text{Total Points}} \right) \]
- Accuracy: The more points you generate, the closer your estimation will be to the actual value of Pi (~3.141592).
Enjoy exploring the mathematical beauty of Pi through the Monte Carlo Simulation! Dive in, experiment with different numbers of points, and see how your estimations improve with each simulation.
Calculation Steps:
Total Points Generated (\(N\)): 0
Points Inside Circle (\(N_{\text{inside}}\)): 0
Points Outside Circle (\(N_{\text{outside}}\)): 0
Pi Estimation (\(\pi\)): 0
Formula Used:
\[ \pi \approx 4 \times \left( \frac{N_{\text{inside}}}{N} \right) \]