Mathematics 7–10 · Year 9
Buffon's needle: estimating pi by dropping toothpicks on ruled lines
Statistics and probability
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The idea
When toothpicks as long as the line spacing are scattered at random over ruled lines, the long-run proportion that cross a line is 2/π (Buffon's 1777 result, whose proof needs calculus and is taken as given here), so the relative frequency of crossings estimates 2/π and hence pi, and the estimate tightens as drops accumulate.
What you need
- A3 sheet ruled with parallel lines spaced exactly one toothpick length apart (measure the toothpick to the nearest millimetre and rule the lines at that spacing with a set square)
- 15 toothpicks of identical length per pair
- Recording sheet: drops, crossings, running estimate 2 × drops / crossings
How to do it
- Measure one toothpick to the nearest millimetre and rule the sheet at that spacing; check three spacings with the ruler.
- Scatter the 15 toothpicks over the whole ruled area, tossing them gently from about 15 cm with a sweep of the hand or dropping them one at a time over different spots, so they land spread across many lines rather than bunched over one; count the toothpicks that touch or cross a line (ignore any that leave the sheet). Record drops and crossings.
- Repeat until the pair has 150 drops; compute the running estimate of pi after each batch.
- Pool the class (10 pairs gives 1500 drops) and compute the pooled estimate.
- Use Buffon's result, given here rather than derived because its proof needs calculus: for a toothpick as long as the spacing the crossing probability is 2/π, so crossings ÷ drops estimates 2/π and rearranging gives π ≈ 2 × drops ÷ crossings.
What you should see
The crossing probability is 2/π = 0.6366. From one pair's 150 drops the expected crossings are 95.5 with a standard deviation of 5.9, so about two times in three the pair's estimate of pi lands between 2.96 and 3.35; from 1500 pooled drops the crossings are 955 ± 19 and the estimate lands between 3.08 and 3.20 about two times in three. The estimate wanders widely early and settles near 3.14 as drops accumulate, but slowly: its standard deviation is about 2.37 ÷ √(drops), so pinning pi to two decimal places (±0.005) reliably needs hundreds of thousands of drops. A single pooled figure has no spread of its own to compare with, so the spread that is measured is the one across the 10 pair estimates. The learner knows it worked when the pooled estimate is within 0.1 of 3.14 (which happens about nine times in ten) and when the standard deviation of the 10 pair estimates comes out near 2.37 ÷ √150 = 0.19, which is √10 = 3.2 times the pooled estimate's own standard error of 2.37 ÷ √1500 = 0.06; with only 10 estimates that measured spread carries about 24 per cent uncertainty of its own, so closer agreement than that cannot be asked for.
What changes
- What you change
- number of drops
- What you measure
- estimate of pi
- What you keep the same
- spacing equal to toothpick length
- drops from the same height
- drops spread over the whole ruled area
- toothpicks off the sheet not counted as drops
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Pi can only come from circles.
- A hundred drops gives pi to two decimal places.
- The estimate improves at a steady rate with each drop.
Safety card
Hazards
- toothpick points
Controls
- drop from low height
- collect all toothpicks at the end
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Probability A; page read 2026-09-22. acara_v9 is empty by decision, not by omission: no ACARA v9 Year 7 to 10 description covers estimating a constant from the long-run relative frequency of a single event, because AC9M9P02 is confined to events joined by 'and' and 'or', AC9M9P03 to comparing simple events with related compound events, and AC9M7P02, whose repeated-experiment wording fits the method best, is Year 7, two years below this entry; v9 records re-read 2026-09-23MA5-PRO-C-01
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.