Mathematics · Interactive learning companion

Monte Carlo Pi Approximation Lab

Use the area ratio of a full circle inscribed in its surrounding square and random sampling to estimate π.

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📝 Description

Monte Carlo Pi Approximation Lab demonstrates how random sampling can estimate the value of pi. Students observe points inside and outside a circle-square model, connecting geometry, probability, and numerical approximation. It supports statistics, computation, and mathematics enrichment.

📋 Instructions

Run the simulation, increase the number of random points, and observe how the pi estimate changes. Compare accuracy as trials increase.

🎓 Learning Guide: Monte Carlo Pi Approximation Lab

Learning focus: Use the area ratio of a full circle inscribed in its surrounding square and random sampling to estimate π.

Mathematics Interactive learning companion

Learning objectives

Relate the area of a full circle to the area of the square that contains it.
Estimate π using four times the fraction of uniformly sampled square points falling inside the circle.

Concepts and principles

For a circle of radius r inscribed in a square of side 2r, the circle-to-square area ratio is πr²/(2r)²=π/4.
If N points are sampled uniformly in the square and C fall inside the circle, the Monte Carlo estimator is π̂=4C/N.
Random estimates fluctuate; increasing the sample size tends to reduce statistical uncertainty but does not guarantee improvement at every step.

Examples and worked solutions

In a square containing a full inscribed circle, 785 of 1,000 uniformly sampled points land inside the circle. Estimate π.: Worked solution: π̂=4×785/1000=3.14.

Try this investigation

Identify the full circle and its enclosing square in the game and the inside-point and total-point counters.
Generate a small batch of points and estimate π using 4 × (inside count)/(total count).
Generate a larger sample and compare the estimate with the earlier result and the known value of π.
Explain why the result may fluctuate even when the sampling method is correct.

Expected observations and explanations

The fraction of uniformly sampled points inside the inscribed circle should approach the area ratio π/4 as the number of trials becomes large.
Sampling uncertainty typically decreases as trial count grows, but a single new batch can produce a less accurate estimate.

Assessment questions and answers

If the number of random points doubles, must the next π estimate be closer to the true value?: Answer: No. A larger sample generally improves statistical precision on average, but random fluctuations can make an individual estimate less accurate.