Measuring Pi — Archimedes' Method of Squeezing a Circle Between Polygons

Pi is the ratio of any circle's circumference to its diameter — the same number for every circle. You can measure it roughly with a string, but around 250 BC Archimedes found it exactly, with no measuring at all. He trapped a circle between a polygon drawn just inside it and one just outside, then doubled the sides again and again until the two polygons closed in on the circle from both directions. With 96-sided polygons he proved pi lies between 3+10/71 and 3+1/7. This blueprint walks through both the string measurement and Archimedes' rigorous squeeze.

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说明

Understand what pi is

Pi is the ratio of a circle's circumference (the distance around) to its diameter (the distance across). It is the same for every circle, large or small. Its value begins 3.14159… and never ends or repeats.

Measure pi with a string

Wrap a cord snugly around a round object and mark its circumference, then measure straight across for the diameter. Divide circumference by diameter — you will get about 3.14. Try several sizes; the ratio stays the same every time.

所需工具：

Cotton Kitchen String

See the limit of measuring

A cord and ruler give only two or three good digits — small measuring errors spoil the rest. To pin pi down exactly you need geometry, not string. This is the leap Archimedes made.

Trap the circle between two polygons

Draw a circle, then a regular polygon just inside touching it and another just outside enclosing it. The circle's circumference must lie between the two polygon perimeters — smaller than the outer, larger than the inner.

所需工具：

Red Alder Board

Chalk Line

Start with hexagons

Begin with six-sided polygons — the easiest to draw, since a hexagon's side equals the circle's radius. Their perimeters already bracket pi between 3 and about 3.46. Good, but still loose.

Keep doubling the sides

Double the sides: 6 to 12 to 24 to 48 to 96. Each doubling makes both polygons hug the circle more closely, so the inner and outer perimeters squeeze together and the gap that must contain pi shrinks.

Reach Archimedes' bounds

At 96 sides Archimedes proved pi is greater than 3+10/71 (about 3.1408) and less than 3+1/7 (about 3.1429). The true value 3.14159… sits right in that gap — found by pure reasoning, with no ruler touching the circle.

Go as far as you like

The doubling never stops: more sides give more digits of pi, as many as your patience allows. Archimedes' squeeze was the best method known for almost two thousand years, until calculus offered faster ones.