Hong Life CalculatorsUnderstanding the pi experiments calculator — concept and principle
This calculator estimates π with a random experiment: either by scattering points in a square (the Monte Carlo method) or by simulating Buffon's needle on a lined floor. It reports the estimate, the number of hits, the error compared with the true π and the typical error for that many trials. Use it to explore probability, simulation and why random estimates converge slowly.
How it is calculated
- Random points: π ≈ 4 × (points inside the quarter circle) ÷ (all points). Typical error ≈ 4√(p(1 − p)/n) with p = π/4.
- Buffon's needle (length l ≤ line spacing d, x = l/d): π ≈ 2x × throws ÷ crossings. Typical error ≈ √(π²/(2n)·(π/x − 2)).
Random numbers come from the browser's cryptographic generator, so each run gives a slightly different result.
The principle
The quarter circle x² + y² ≤ 1 inside the unit square has area π/4, while the square has area 1. A point chosen uniformly at random lands inside with probability π/4, so the hit fraction estimates π/4. For Buffon's needle, a needle shorter than the line spacing crosses a line with probability 2l/(πd), first worked out by Georges-Louis Leclerc, Comte de Buffon, in the 18th century. In both cases the estimate is a proportion, and the error of a proportion shrinks like 1/√n: a hundred times more trials gives only about ten times more accuracy.
Worked example
Default inputs: random points, 100,000 trials. The run shown on the page gave:
- Hits inside the quarter circle: 78,593.
- Hit fraction: 78,593 ÷ 100,000 = 0.78593.
- Estimate: 4 × 0.78593 = 3.1437 (3.14372).
- Difference from π: 3.14372 − 3.14159 = 0.002127, or 0.07%.
- Typical error: 4√(0.7854 × 0.2146 ÷ 100,000) ≈ 0.005193.
Your run will show different hits, but the estimate should usually fall within about ±0.005 of π.
| Number of trials (points or needle throws) | Estimate of π |
|---|---|
| 25,000 | 3.1304 |
| 50,000 | 3.1312 |
| 75,000 | 3.1354 |
| 100,000 | 3.1355 |
| 125,000 | 3.1464 |
| 150,000 | 3.1385 |
| 200,000 | 3.1411 |
How to read the result
Compare the difference from π with the typical error. A difference smaller than one typical error is normal; about two times larger happens around 5% of the time. To gain one more correct decimal you need roughly 100 times as many trials. A long needle ratio (close to 1) gives a lower error for Buffon's method.
Common mistakes
- Expecting many correct decimals: a million points usually gives only 2 to 3.
- Judging the method from one lucky or unlucky run.
- Using a needle longer than the line spacing, where the simple formula no longer applies.
Frequently asked questions
Why multiply by 4?
The quarter circle covers π/4 of the square, so the hit fraction must be multiplied by 4 to get π.
Why does my result change every time?
The points are random. Each run is a new sample, so the estimate fluctuates around π.
How many trials give 3.14 reliably?
Roughly a few hundred thousand points; the typical error at 100,000 is about 0.005.
Sources: Weisstein, E. W., "Buffon's Needle Problem", Wolfram MathWorld