How Did Archimedes Calculate Pi Without a Calculator?

Imagine a world with no calculator. You draw a circle. Then someone asks, “How long is the curved edge?”

A straight line is easy to measure. A curved line is harder. More than 2,000 years ago, Archimedes found a clever way around this problem.

Quick Answer

Archimedes used polygons, which are shapes made with straight sides. He put one polygon inside a circle and another outside it.

The inside polygon was too short. The outside polygon was too long. So the real circle had to sit between them.

Archimedes did not guess one number. He built two fences around the answer.

First, What Is Pi?

Take any circle. Measure the distance around it. This distance is called the circumference.

Now measure straight across the circle through its center. This is the diameter.

Divide the circumference by the diameter. You get the same special number for every circle. We call that number pi, or π.

circumference ÷ diameter = π

Today, we often write π as 3.14159… But Archimedes did not have a calculator or modern decimal notation.

The Hard Part: A Circle Has a Curved Edge

Suppose you want to measure a square. Its sides are straight. You can measure each side and add the lengths.

A circle is different. Its edge keeps turning. How can you measure that curve with straight-line geometry?

Archimedes changed the problem. Instead of measuring the curve directly, he used shapes made from straight lines.

Put a Polygon Inside the Circle

Start with a regular polygon inside the circle. Its corners touch the circle.

Because the polygon cuts across the curved edge, its perimeter is shorter than the circle.

That gives us a number that is too small.

Now Put Another Polygon Outside

Next, place a regular polygon around the circle. Each side touches the circle from the outside.

This outside path is longer than the circle. So it gives us a number that is too large.

Now the circle is trapped.

Think before you read on.

What should happen if both polygons get more sides? Will the gap get bigger or smaller?

Add More and More Sides

Archimedes kept making the polygons finer. A standard reconstruction follows this sequence:

6 → 12 → 24 → 48 → 96 sides

Figure 1. More sides make the straight edges follow the circle more closely.

Watch what happens.

• The inside polygon gets longer.
• The outside polygon gets shorter.
• The circle stays between them.
• The gap gets smaller.

This is the big idea. A hard curved problem can be squeezed between two easier straight-sided problems.

How Close Did Archimedes Get?

In Measurement of a Circle, Archimedes gave a lower and an upper bound for the circle ratio.^[1]

223/71 < π < 22/7

In decimal form, those two fences are about:

3.1408 < π < 3.1429

The true value starts 3.14159… So Archimedes had trapped it inside a very small space.

Does Pi Equal 22/7?

No.

This is an important point. Archimedes did not say that π equals 22/7.

He showed that π is less than 22/7. The fraction was the upper fence, not the exact answer.

22/7 is close to pi, but it is not pi.

Why This Method Matters

Archimedes did something deeper than find a useful number. He showed a way to think.

When a problem is hard, we can replace it with simpler pieces. Then we can make those pieces finer and compare the results.

Much later, mathematics gave new names and tools to ideas about getting closer and closer. Modern computers also break difficult shapes and calculations into many simpler parts.

The modern methods are much more advanced. They are not the same as Archimedes’ method. But the problem-solving habit is still powerful.

A Modern Way to Recreate the Idea

Today, a short Python program can repeat the same geometric idea with modern trigonometry.

You only change the number of sides. The computer then shows a lower and an upper estimate.

Python — modern reconstruction

import math

n = 96

lower = n * math.sin(math.pi / n)
upper = n * math.tan(math.pi / n)

print("lower:", lower)
print("pi:   ", math.pi)
print("upper:", upper)

This code uses modern sine and tangent functions. Archimedes did not use this notation. His original reasoning was geometric.^[2]

Try this.

Change n = 96 to 6, 12, 24, or 48. What happens to the gap?

Then try 192. Can you predict the result before you run the code?

Words to Keep

circumference
The distance around a circle.

diameter
A straight line across a circle through its center.

polygon
A closed shape made with straight sides.

estimate
A value that is close to the true value.

bound
A value that the answer cannot pass.

One Sentence to Keep

Archimedes did not measure the curve directly. He trapped it between shapes he could calculate.

What Should We Ask Next?

A 96-sided polygon is still a polygon. It is not a circle.

So here is our next question:

What happens if we keep adding more and more sides?

That question will lead us toward one of the biggest ideas in mathematics: getting closer and closer without simply jumping to the answer.

Sources & Further Reading

1. Archimedes, Measurement of a Circle, in The Works of Archimedes — edited and translated by Thomas L. Heath, Cambridge University Press.
2. A History of Pi — MacTutor History of Mathematics, University of St Andrews. This source notes that modern trigonometric notation is an unhistorical reconstruction of Archimedes’ geometry.
3. G. M. Phillips, “Archimedes the Numerical Analyst” — The American Mathematical Monthly, 88(3), 1981, pp. 165–169.
4. David Weisbart, “Modernizing Archimedes’ Construction of π” — Mathematics, 8(12), 2204, 2020.

Historical note: modern formulas and Python code are used here to recreate the geometric idea for today’s learner. They are not presented as Archimedes’ original notation.