How did Archimedes trap pi between two polygons?
With nothing but straight-sided polygons, Archimedes proved that pi lies between 223/71 and 22/7.
▶ Start the storyArchimedes trapped pi between two polygons. He did not try to measure a circle directly. Instead he squeezed it. A regular polygon drawn inside a circle has a perimeter that is less than the circle's circumference, and a polygon drawn around the outside has a perimeter that is greater. The ratio of circumference to diameter, which we call pi, must sit between the two.
He started with hexagons, inside and outside, whose perimeters are easy to work out. (A hexagon inside a circle has a perimeter of exactly three diameters, so the very first step already shows that pi is more than 3.) Then he showed how to find the perimeters when the number of sides is doubled, and kept doubling: 12, 24, 48, 96. With 96-gons he proved that pi is bigger than 223/71 and smaller than 22/7, in the 3rd century BCE.
Step 1: A circle with its diameter
Pi is circumference divided by diameter
Step 2: Hexagons inside and outside
Perimeters are easy to find
Step 3: Double the sides
12, then 24, then 48
Step 4: 96 sides
223/71 < π < 22/7
Step 5: Inner perimeter < circle < outer perimeter
The two bounds squeeze pi
The hard part wasn't the geometry but the arithmetic. He used no trigonometry, and each doubling needs a square root, which he had to approximate with fractions. How he found them he never explained. Nobody knows why he stopped at 96 sides, either: it only takes patience to go further.
Pi had been approximated before. Babylonian mathematics usually approximated it as 3, which was good enough for the architectural projects of the time. What Archimedes added was a method that comes with guaranteed bounds, and could, in principle, be pushed as far as you liked. Later mathematicians did push the method with more sides: Liu Hui in China used a 96-gon and a 192-gon, and around 1600 Ludolph van Ceulen computed 35 decimal places and had them engraved on his tombstone.
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Recap
Pi lies between the perimeter of a polygon inside the circle and one outside, and doubling the sides squeezes the gap.
💡 A trick to remember it · One polygon inside, one outside: the circle is the meat of a sandwich whose bread keeps getting thinner.
Surprising fact · Nobody knows why Archimedes stopped at 96 sides.
Connects to
- 📏 Why is a 2,300-year-old geometry book called the most successful textbook ever?
- 📈 How did Descartes turn shapes into equations?
- 📐 Why did two mathematicians both claim to have invented calculus?
- 📜 How was a lost book by Archimedes found hidden under a prayer book?
- 🥧 What is pi, and why do its digits never end?
Sources (2)
No source, no claim. Every fact in this lesson (15 claims) cites at least one of these.