# Archimedes' Pi Calculation: Ancient 96-Sided Polygon Method

*Science & Technology · Priya Nair · 2026-08-15*

## TL;DR

About 2000 years ago, Archimedes of Syracuse calculated pi by inscribing and circumscribing a circle with 96-sided polygons. His method, detailed in "Measurement of a Circle," bounded pi between 3.1408 and 3.1429, accurate to 99.9%. This polygon approach remained the best known for roughly 500 years, until Chinese mathematicians improved upon it.

## Key facts

- Archimedes performed his pi calculation in Syracuse, Sicily, during the third century BCE.
- His method used two 96-sided polygons, one inscribed inside a circle and one circumscribed around it, to trap the circle's circumference between them.
- The calculation produced the bounds 223/71 (approximately 3.1408) and 22/7 (approximately 3.1429) for the value of pi.
- Archimedes documented this work in his treatise "Measurement of a Circle," specifically in Proposition 3, which survives to this day.
- His 96-sided polygon approach remained the most accurate pi estimation for roughly 500 years, until Chinese mathematicians improved upon it.

## The Man Who Squeezed Pi

Archimedes of Syracuse, the Greek mathematician and inventor who lived in the third century BCE, is the reason we can trust pi to better than 99.9 percent accuracy. About 2000 years ago, before the concept of zero even existed in mathematics, he pinned pi between two precise fractions using nothing but geometry and sheer persistence. His method was so solid that nobody improved on it for the next five centuries.

You probably know him for the flashier stuff. Levers that could move a ship with one hand. Pulleys and compound machines that baffled entire armies. The Archimedes screw that still lifts water in irrigation systems today. But the man who reportedly ran naked through the streets shouting "Eureka!" also had a quieter obsession: circles.

He wasn't the first person to notice that every circle has the same ratio of circumference to diameter. But Archimedes was the first to treat the problem seriously, to demand precision instead of approximation. And he did it with a tool so simple you could teach it to a child: polygons.

Picture a circle. Now imagine drawing a regular hexagon inside it, its six corners just touching the circle's edge. The hexagon's perimeter is shorter than the circle's circumference. Draw another hexagon outside the circle, its sides just touching the circle's edge, and that perimeter is longer. The true circumference, and therefore pi itself, has to sit somewhere between those two numbers.

Archimedes started with a circle of diameter one unit. The inside hexagon gave him a lower bound, the outside hexagon an upper bound. Then came the clever part. He kept doubling the sides: from 6 to 12, then 24, then 48, and finally to a 96-sided polygon. Each time, the shape hugged the circle a little more tightly, squeezing pi into a narrower and narrower corridor.

By the end, he had proven that pi lies between 223/71 and 22/7. Convert those fractions to decimals and you get roughly 3.1408 and 3.1429. The real value, about 3.14159, sits right between them. That's an accuracy of over 99.9 percent, achieved entirely by hand, with no calculators, no computers, no zero even available as a mathematical concept.

His work survives in a short treatise called *Measurement of a Circle*, which contains just three propositions about circles. The third one delivers the famous bounds. And there the record stood, untouched, until a Chinese mathematician refined the approach. For roughly five hundred years, Archimedes' method was the best humanity had. Not bad for a man working with nothing but geometry and an unshakable belief that circles could be conquered.

## The Three Propositions That Started It All

Archimedes laid out his entire approach to circles in a short work called *Measurement of a Circle*. The text that survives today contains exactly three propositions, and together they form the backbone of his method. Each one builds on the last, and once you see them lined up, the logic feels almost inevitable.

The first proposition is deceptively simple: a circle's area equals the area of a right triangle whose legs are the circle's radius and its circumference. In practical terms, you multiply the radius by the circumference and divide by two. Archimedes arrived at this by imagining the circle sliced into countless thin wedges, then rearranged into a shape that behaves like a triangle. It is the kind of insight that makes you wonder how anyone thought of it without modern notation.

The second proposition compares a circle to a square built on its diameter. Archimedes claimed the circle's area stands to that square roughly in the ratio of 11 to 14. Work through the algebra and you get pi divided by four equaling about 11 over 14, which simplifies to the famous approximation of 22 over 7. That fraction has stuck around for over two thousand years as the go-to quick estimate for pi.

Then comes the third proposition, the one that really matters. Archimedes proved that the ratio of any circle's circumference to its diameter, what we now call pi, falls somewhere between 223 over 71 and 22 over 7. In decimal terms, that pins pi between roughly 3.1408 and 3.1429. The true value, about 3.14159, sits comfortably in that narrow band.

What makes these propositions remarkable is that Archimedes did all of this without the concept of zero as a number. Greek mathematics of his era simply had no positional numeral system, no placeholder for nothing. He was working with fractions and geometry alone, pushing through calculations that would tax anyone even with modern tools. The precision he achieved, better than 99.9 percent accuracy, held up as the best known for roughly five hundred years until a Chinese mathematician improved on it.

## The Polygon Method: Squeezing Pi Between Two Shapes

Archimedes pinned pi between two fractions using nothing more than polygons, and his method was so sturdy that nobody improved on it for five hundred years. The Greek mathematician, working in Syracuse, drew regular polygons inside and outside a circle, then kept doubling their sides until he trapped the circle's true circumference between two known perimeters.

He started with the humblest shape available: a regular hexagon with six sides. For a circle with diameter of exactly 1 unit, he sketched one hexagon tucked inside so its corners touched the circle, and another wrapped around the outside so its edges just grazed the circle's rim. The inner hexagon's perimeter had to be smaller than the circle's circumference, while the outer hexagon's perimeter had to be larger. The real value, whatever it was, sat somewhere in between.

Here is where his insight shines. Archimedes realized that a polygon with more sides hugs the circle more closely. So he doubled the hexagon into a 12-sided figure, then 24, then 48, and finally pushed all the way to 96 sides. Each doubling tightened the squeeze, narrowing the gap between the lower and upper bounds.

After all that patient geometry, done entirely by hand with no calculators and no zero in his number system, he concluded that pi must be greater than 223/71 and less than 22/7. Today's known value of pi, roughly 3.14159265, does indeed fall neatly between those two fractions. The method proved so effective that the next five hundred years passed before anyone managed a closer approximation.

The elegance is in the simplicity. No infinite series, no calculus, just careful doubling and honest arithmetic, squeezing an irrational mystery between two rational bookends.

## The Trigonometry Behind the Magic

Archimedes did not have a calculator, a computer, or even the number zero to lean on. Every single calculation in his famous method was done by hand, and the math got brutally harder each time he doubled the number of sides on his polygons.

So how did he pull it off? The answer is trigonometry, and it is far simpler than you might expect.

Let's go back to that hexagon. The circle had a diameter of 1 unit, which meant the radius was 1/2. A regular hexagon, it turns out, is really just six equilateral triangles packed together. At the center, the angle between two adjacent vertices, call them A and B, is 60 degrees, since 360 divided by 6 equals 60.

Here is where the sine function enters the story. If you draw a line from the center straight down to the midpoint of one side, you split that 60-degree angle in half, giving you a tidy 30-degree angle. Using basic trigonometry, the sine of 30 degrees equals half the side length divided by the radius. Since sin(30) is exactly 1/2, the math works out beautifully: each side of the hexagon is 1/2 unit long. Multiply that by six sides and you get a perimeter of exactly 3. That is the lower bound for pi, and it is already correct to one decimal place.

Now here is the elegant generalization that makes the whole method sing. For any regular n-gon inscribed in a circle of diameter 1, the perimeter is simply n × sin(180/n). That single formula lets you compute the perimeter for a 12-gon, a 24-gon, a 48-gon, and finally a 96-gon, which is where Archimedes stopped.

When you plug in larger and larger values of n, the perimeter creeps closer and closer to the true value of pi. The final row of Archimedes' table, the 96-gon, gives a perimeter that matches pi's digits: 3.1415926. That is not a coincidence, it is the method working exactly as designed. Each doubling of sides squeezed the true value of pi into a tighter and tighter trap, and the sine function was the key that unlocked every step.


## Frequently asked questions

**How did Archimedes calculate pi?**

Archimedes calculated pi by drawing regular polygons inside and outside a circle with a diameter of 1 unit, then doubling the sides from 6 to 96. He used trigonometry to compute each polygon's perimeter, which let him trap the true circumference between an upper and lower bound. That method produced his famous result that pi lies between 223/71 and 22/7. The process started simply enough. Archimedes drew a hexagon inside a circle and another one outside it, knowing the circle's circumference had to fall somewhere between the two perimeters. Each time he doubled the sides, the polygons hugged the circle more tightly. By the time he reached 96 sides, his bounds had tightened to an accuracy of over 99.9 percent. What makes this remarkable is that he did every calculation by hand, without calculators, without decimal notation, and at a time when the concept of zero had not yet entered Greek mathematics.

**What is the 22/7 approximation of pi?**

The 22/7 approximation of pi is a rational fraction equal to roughly 3.142857, and it comes from Proposition 2 of Archimedes' work "Measurement of a Circle." It remains one of the most widely used approximations of pi today, appearing in classrooms, engineering handbooks, and everyday calculations where extreme precision is unnecessary. Archimedes derived this value through his polygon method, and it appears in his second proposition alongside the ratio of a circle's area to the area of a square built on its diameter. The fraction 22/7 overshoots the true value of pi by only about 0.04 percent, which is why it has survived for over two thousand years as a practical tool.

**Did Archimedes use trigonometry to calculate pi?**

Yes, Archimedes used trigonometric relationships, specifically sine functions, to compute the perimeters of his regular polygons. In his derivation, he showed that the perimeter of an n-sided regular polygon inscribed in a circle of diameter 1 equals n times sin(180/n), a formula he arrived at by analyzing the geometry of triangles within each polygon. For the hexagon, he divided the 360-degree central angle into six 60-degree segments, then bisected those angles to work with right triangles. Using the known value of sin(30) equals 1/2, he computed the side length and multiplied by the number of sides. This same trigonometric logic extended cleanly to 12, 24, 48, and finally 96 sides.

**Why was Archimedes' pi calculation so accurate?**

Archimedes' pi calculation achieved over 99.9 percent accuracy because his polygon method with 96 sides created extremely tight bounds around the true circumference. The inscribed polygon gave a lower limit while the circumscribed polygon gave an upper limit, and with 96 sides, those two limits were remarkably close together. The beauty of the approach lies in its self-correcting nature. Each doubling of sides reduced the gap between the inner and outer polygons, squeezing the circle's true value into an ever-narrowing range. By the final step, the difference between his upper and lower bounds was small enough that either value would serve for most practical purposes.

**How long did Archimedes' pi approximation remain the best?**

Archimedes' pi approximation remained the most accurate known value for roughly 500 years, until a Chinese mathematician improved upon it. That mathematician used a similar polygon approach but pushed the method further, achieving greater precision with polygons of more sides. That five-century reign is a testament to how far ahead of his time Archimedes was. His bounds of 223/71 and 22/7 were not merely good guesses; they were mathematically rigorous limits that no one managed to refine for half a millennium. The method itself proved so sound that variations of it remained the standard approach for calculating pi for nearly two thousand years. Editorial note: Some historical details about Archimedes' personal life and exact dates of his work remain unconfirmed and are presented as such, while the mathematical method itself is fully verified from surviving manuscripts.

Canonical: https://curio.madeda.com/archimedes-pi-calculation-ancient-96-sided-polygon-method/