Buffon's Needle - Dropping Sticks to Find Pi

Estimated Value of Pi

—

Drop needles to begin

Drop Speed

50 needles/sec

Quick Drop

Total Needles 0

Crossing Lines 0

Hit Rate —

Actual Pi 3.14159265...

The Formula

π ≈ 2n / h

where n = total drops, h = hits

"In 1901, Lazzarini dropped 3,408 needles and calculated Pi to 6 decimal places!"

— Mario Lazzarini's famous experiment

📐 The Setup

Draw parallel lines on a floor, spaced d units apart. Drop a needle of length l (where l ≤ d). Count how many needles cross a line. The probability of crossing is 2l/(πd). Rearranging: π = 2ln / (dh).

🎲 Why Does This Work?

A needle crosses a line if its center's distance to the nearest line is less than ½l × sin(θ), where θ is the angle. Integrating over all angles (0 to π) and all distances (0 to d/2) gives the probability formula involving pi—it appears naturally from the geometry!

📊 Monte Carlo Method

This is one of history's first Monte Carlo simulations— using randomness to estimate mathematical constants. The same principle powers modern computer graphics, financial modeling, and nuclear physics calculations. Random sampling converges to truth!

🏛️ Historical Note

Proposed by Georges-Louis Leclerc, Comte de Buffon in 1777. He was a naturalist who also estimated Earth's age at 75,000 years (bold for his time!). The needle problem was his venture into probability theory—with spectacularly beautiful results.