Cutting the Circle — Liu Hui and the road to π
No curved ruler? No problem. Trap the circle between straight lines and squeeze — the way a Chinese mathematician did in 263 CE.
How can you measure a circle using only straight lines? You cheat gently: draw a polygon inside the circle and keep doubling its sides. The perimeter of the polygon is easy to compute, and as the sides multiply, the polygon hugs the circle ever closer, so its perimeter creeps towards the circumference. Around 263 CE the Chinese mathematician Liu Hui pushed this to a 3,072-sided polygon and pinned π at 3.1416 — without modern algebra.
| Sides | One side | Perimeter | π ≈ | Misses π by |
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The circle's radius is 1, so its true circumference is exactly 2π ≈ 6.2832. Half the polygon's perimeter is therefore an estimate of π — computed live from the geometry (side = 2·sin(180°/n)), never typed in.
Nearly 1,800 years ago the Chinese mathematician Liu Hui (around 263 CE) computed π exactly this way — "cutting the circle" (割圆术). He started from a hexagon and doubled the sides again and again, all the way to a 3,072-sided polygon, reaching π ≈ 3.1416 with nothing but a ruler, counting rods and patience. His rule: "The finer you cut, the less is lost; cut and cut again until it cannot be cut — then it coincides with the circle and nothing is lost." Every number in the table above is his method, recomputed in your browser.
How to play
Use the slider or the Double the sides button to run through 6, 12, 24, 48 and 96 sides. The table shows each perimeter and the π estimate it produces, with the shrinking error beside it. Watch the drawing as the polygon melts into the circle.
What you learn
You meet one of the deepest ideas in mathematics: an exact value you cannot reach directly can still be approached as closely as you like, by better and better approximations. π stops being a magic constant and becomes the end of a road you can walk.
Want more? Browse the whole Interactive Geometry Lab for nine hands-on math toys, or build a personalised math adventure for your child with the free Game Builder. Everything is free, with no sign-up.
✓ Last checked November 2, 2026 — we re-test this tool every week to make sure it still works.