Why π turns up where there are no circles

Drop a needle onto a floor of parallel planks. Count how often it crosses a crack. The proportion is 2/π when the needle is as long as a plank is wide. No circle was drawn, and π appears anyway.

Why it is cool

Buffon posed the problem in 1777, and it is the earliest example of estimating a mathematical constant by running a random experiment — the method now called Monte Carlo. It also settles a common confusion. π is not a property of circles; it is a property of angles. Wherever a direction is chosen at random, π is already in the room.

The mechanism

Let the planks be width 1 and the needle length 1. Where the needle lands is described by two numbers: the distance d from its centre to the nearest crack, which is anywhere from 0 to ½, and the angle θ it makes with the cracks, which is anywhere from 0 to a right angle. Both are equally likely across their ranges.

The needle reaches a distance of ½·sin θ from its centre toward the crack. It crosses when that reach is at least d: when d ≤ ½·sin θ.

Now draw the picture. Put θ across the page from 0 to a right angle, and d up the page from 0 to ½. Every landing is a point in this rectangle, and every point is equally likely. The crossings are the points under the curve d = ½·sin θ. So the crossing probability is the area under half a sine hump divided by the area of the rectangle.

The rectangle has area (π/2)·(½) = π/4, because a right angle is π/2 radians. The area under ½·sin θ from 0 to π/2 is ½ — sine's hump from 0 to a right angle encloses exactly one unit of area, and we have half of it. Divide: (½)/(π/4) = 2/π ≈ 0.6366.

That is where the circle went. Measuring an angle in radians means measuring it as a length around a unit circle, so a right angle is π/2. Every random angle carries π with it.

In one breath

A needle crosses a crack when its centre is closer than half its sideways reach, and the reach depends on a random angle. Averaging over all angles gives an area under a sine curve divided by a rectangle whose width is a right angle in radians. That ratio is 2/π. Count crossings, invert, and π falls out of a floor.