Key Concepts: Monte Carlo Methods in Geometry ⭐

What Is Monte Carlo?

A computational technique that uses random sampling to estimate geometric quantities like areas or volumes.

Estimating π

• Inscribe a circle of radius r inside a square of side 2r.
• Generate random points in the square.
• Count how many fall inside the circle.
• π ≈ 4 × (points inside circle) / (total points).

Estimating Irregular Areas

Enclose the region in a rectangle. Generate random points. The area ≈ (hits inside region / total points) × area of rectangle.

Accuracy

More random samples → better estimate. Error decreases proportionally to 1/√n, where n is the number of samples.