Chinese Pi Calculations: From Liu Hui to Zu Chongzhi

An old recipe for a circular field and a modern search for digits of π can pose the same mathematical problem. They put very different demands on the answer. The history of Chinese pi calculations shows when a simple multiplier worked and how mathematicians checked its limits.

Why Older Chinese Problems Used Three for Pi

The opening chapter of The Nine Chapters on the Mathematical Art (九章算术, Jiǔzhāng Suànshù) contains practical problems about measuring land. Its circle rules were compatible with a circumference three times the diameter. For a round field 20 units across, this gives 60 units around the edge. Using a more precise value of π gives about 62.83 units. These are modern sample numbers to show the size of the difference.^[1]

The more informative formula uses half the circumference multiplied by half the diameter. If you already know the circumference and diameter, their product divided by four yields the area without having to substitute a value of π at that step. A rule expressed entirely in terms of diameter has to build in an approximation. The choice changes the answer by a predictable amount.^[1]

This is one reason to read the text as a collection of procedures. Ask which quantities the problem gives and which the rule has to infer. That distinction carries over to Chinese surveying mathematics, where the quantities a surveyor can measure determine the calculation that follows.^[3]

How Liu Hui Cut a Circle into Polygons

In his commentary dated 263 CE, Liu began with a regular hexagon inside a circle. A line joining two neighboring corners lies inside the curved edge. Add a corner on each arc to turn the hexagon into a 12-sided polygon, then repeat the doubling. The straight-sided figure fills progressively more of the circle while leaving progressively smaller gaps.^[1][3]

The calculation works with area. Join the center to the polygon’s corners and think of the area as a sum of triangles. Each doubling uses the geometry of the previous figure to find the new side length and the area added. The inscribed polygon gave a lower bound for the circle’s area. Liu then used the difference between successive polygons to help bound the area still left outside. The gougu theorem explains the right-triangle relationship used in finding lengths in this construction.^[1]

The old value of 3 makes the scale of the improvement easy to see. For a circle of radius one, its area in square units equals π. Three falls short of the area of the circle, and 3.14 comes much closer. Liu kept the simpler ratio 157/50 available for ordinary work. The written commentary matters because readers can follow how the polygons were made and how the successive approximations were compared.^[1][3]

What Survives of Zu Chongzhi’s Calculation

The History of the Sui (隋书, Suíshū), compiled after Zu’s lifetime, records that he placed π between 3.1415926 and 3.1415927. It also gives two ratios of circumference to diameter, 355 to 113 and 22 to 7, as the closer and rougher choices respectively. The interval and the fractions survive together in the historical account.^[1][2]

Zu’s mathematical book survives only by report, so the same evidence does not preserve his sequence of operations. Extending Liu’s approach is a plausible reconstruction. Different modern accounts propose different polygon sizes or ways to reach the closer fraction. Even when a reconstruction reproduces the result, it cannot by itself establish which calculation Zu performed. The guide to primary and secondary historical sources explains how a later record can preserve an earlier result while leaving its derivation uncertain.^[1][2]

Why 355/113 and the Decimal Bounds Tell Different Stories

The difference between the two recorded endpoints is one ten-millionth. Truncate any value strictly inside that interval to seven decimal places and you get 3.1415926. By contrast, dividing 355 by 113 gives 3.1415929204 to ten decimal places. It lies just above the upper endpoint. That arithmetic is consistent with the historical account because the fraction is an approximation and the interval is a claim about where the real value lies.^[1][2]

For a diameter of 100 units, the rough fraction 22/7 overshoots the modern circumference by about 0.126 units. The closer fraction 355/113 overshoots by about 0.0000267 units. This is a modern calculation using the two historical ratios. It shows the practical attraction of the closer fraction. A short division gives nearly the same circumference as using many decimal digits of π.^[1][2]

FAQs About Chinese Pi Calculations

Did Liu Hui calculate pi as 3.1416?

The transmitted commentary includes 3927/1250, or 3.1416, alongside discussion of a 3,072-sided polygon. Some historians attribute that passage to Liu. Others question whether the later computation was his. The method and simpler 157/50 ratio have firmer attribution.^[1][3]

Did Zu Chongzhi invent 22/7?

The History of the Sui presents 22/7 as Zu’s rougher working ratio. The same fraction was known independently, and historians do not generally treat its discovery as Zu’s achievement. His distinctive results are the narrow interval and the much closer 355/113.^[1][2]

Sources

[1] Lam Lay-Yong and Ang Tian-Se, Historia Mathematica. Circle Measurements in Ancient China

[2] University of St Andrews, MacTutor History of Mathematics. Zu Chongzhi

[3] University of St Andrews, MacTutor History of Mathematics. Chinese Mathematics Overview