Drop needles at random positions and angles onto a floor ruled with evenly spaced parallel lines. The probability a needle crosses a line depends directly on π, so counting crossings lets you estimate π from pure geometry and chance.
This tool recreates Buffon's needle, the 1733 probability puzzle in which needles of length l are dropped onto a floor ruled with parallel lines spaced d apart. Each needle lands at a uniformly random position and angle, and whether it crosses a line depends on simple trigonometry. Because the theoretical crossing probability is 2l/(πd), tallying dropped needles against crossings and rearranging the formula produces a live, ever-refining estimate of π.
Each needle gets a random centre position y modulo d and a random angle θ in [0, π). It crosses a line when (l/2)·sin(θ) is at least the needle centre's distance to the nearest ruled line. Since P(crossing) = 2l/(πd), the running ratio of needles to crossings gives π ≈ (2·l·N)/(d·C).
Adjust the needle-length slider to change l relative to the fixed line spacing d (kept ≤ d), and the drop-rate slider to control how many needles fall per frame. Use Pause to freeze the animation and Reset to clear all needles and begin a fresh trial. The stats box tracks needles dropped, crossings and the current π estimate with its error.
Buffon's needle is considered one of the very first Monte Carlo methods, predating the formal theory by roughly two centuries. It shows that a purely geometric probability experiment — with no explicit circles at all — can still reveal π, because the sine function connecting angle to crossing distance carries π implicitly.
Buffon's needle problem, posed in 1733, asks: if a needle of length l is dropped at random onto a floor ruled with parallel lines spaced d apart, what is the probability the needle crosses a line? When l ≤ d, the answer is P = 2l/(πd), linking a simple physical experiment directly to π.
Since the crossing probability equals 2l/(πd), observing the fraction of dropped needles that cross a line and rearranging the formula isolates π: π ≈ (2·l·N)/(d·C), where N is the number of needles dropped and C is the number of crossings. More drops produce a more reliable estimate.
The needle length l relative to the line spacing d sets the base crossing probability. Longer needles (closer to l = d) cross lines more often, giving more crossing events per drop and a statistically more stable estimate for the same number of throws, though the underlying formula remains valid for any l ≤ d.
Yes. It uses repeated random trials — random needle position and angle — to estimate a mathematical constant from an observed frequency, which is exactly the definition of a Monte Carlo method. It predates the formal Monte Carlo technique developed at Los Alamos in the 1940s by roughly two centuries.
Like all Monte Carlo estimates, Buffon's needle converges at a rate proportional to 1/√N, so quadrupling the number of drops only halves the typical error. Reaching high precision this way requires an enormous number of needles, which is why claimed high-precision historical results are viewed with scepticism.