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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.
中级
2
说明
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1
Understand what pi is
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.
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2
Measure pi with a string
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 String3
3
See the limit of measuring
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.
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Trap the circle between two polygons
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 Line5
5
Start with hexagons
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.
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Keep doubling the sides
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.
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Reach Archimedes' bounds
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.
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Go as far as you like
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.
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