九章算術
Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024 (a low estimate of the number). Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle. Zu Chongzhi later discovered the calculation of pi to be 3.1415926 &lt; π &lt; 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century.