Ten Famous Pi Approximations: From Ancient Mesopotamia to the 18th Century

Pi is irrational and transcendental — it never ends, never repeats. Yet it appears everywhere. You simply cannot do without it. That is why, for thousands of years, people have had to rely on approximations in everyday life. Here are 10 of the most well-known pi approximations, ordered from least to most accurate.

The Roughest Pi Approximations: Ancient Mesopotamia and the Number 3

10. The value: 3
Mesopotamia.
Ancient peoples had a great fondness for whole numbers and mental arithmetic. It sounds crude, but it was perfectly convenient for measuring the walls of Jericho.
Error: +4.5%

9. √10 ≈ 3.162
Ancient India.
A decent approximation for when you have no calculator but still want to look clever. Remarkably, √10 is very close to π — coincidence?
Error: +0.65%

Egyptian and Babylonian Pi Approximations

8. 256 / 81 ≈ 3.1605
Egypt, Rhind Papyrus.
An ingenious construction based on the diameter. Brilliant for its era.
Error: +0.6%

7. 25 / 8 = 3.125
Babylon.
A fraction conveniently written in the sexagesimal number system.
Error: −0.5%

Archimedes and the Most Famous Pi Approximation

6. 22 / 7 ≈ 3.142857
Archimedes, 3rd century BC.
The most famous pi approximation of all time. This is precisely the fraction celebrated on Pi Approximation Day, observed on July 22nd.
Error: +0.04%

5. 223 / 71 ≈ 3.140845
Archimedes.
The lower bound of π, derived using 96-sided polygons. Remarkably precise.
Error: −0.02%

Chinese Pi Approximations: Precision Without Machines

4. 333 / 106 ≈ 3.141509
China, 1st–4th centuries.
Easy to compute, accurate to 5 decimal places. Even today, a powerful option for mental calculations.
Error: −0.0027%

3. 355 / 113 ≈ 3.1415929
Zu Chongzhi, 5th century.
The Chinese mathematician achieved the seemingly impossible: 7 digits of accuracy with no computing machines. A record that stood for 1,000 years.
Error: +0.0000085%

The Most Precise Historical Pi Approximations: 17th and 18th Centuries

2. 103993 / 33102 ≈ 3.14159265301
Europe, 17th century.
An advanced continued fraction. Appropriate for 8-bit microcontrollers and mathematics olympiads alike.
Error: −0.0000000184%

1. 104348 / 33215 ≈ 3.14159265392
18th century and beyond.
The formula for those who want to compress π into 16 bits without losing precision. Brilliant in its compactness.
Error: +0.0000000105%

Which pi approximation is your favorite?