How were fields surveyed using mathematical procedures in The Nine Chapters on the Mathematical Art?

In The Nine Chapters on the Mathematical Art, the surveying of fields was addressed through the application of geometric algorithms designed to calculate areas. These procedures were presented as lists of operations to be performed on specific data, reflecting the practical needs of civil administration, such as land management and taxation.12

The text provided formulas for various elementary plane figures, including triangles, rectangles, trapezoids, and circles, as well as segments of circles. For instance, to determine the area of a circle, the work provided algorithms that evolved over time; while early versions used an approximation of 3 for $\pi$, later commentaries—most notably by Liu Hui—refined these calculations and provided proofs for more accurate formulas, such as multiplying half the diameter by half the circumference. By organizing these mathematical achievements into a systematic format of problems and general solution procedures, The Nine Chapters established a foundational approach to geometry that allowed officials to measure land and resources effectively.21