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How did Liu Hui prove the formula for the area of a circle?

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East Asian mathematics Counting boards and markers, or counting rods, were used in China to solve systems of linear equations. This is an example from the 1st century ce.

In his 3rd-century CE commentary on The Nine Chapters on the Mathematical Art, the Chinese mathematician Liu Hui proved the formula for the area of a circle—which states that the area is equal to half the diameter multiplied by half the circumference—by using a method of dissection.12

Rather than relying on static geometric theorems, Liu Hui established the correctness of this algorithm by conceptually dividing the circular region into infinitely many pieces. By dissecting the circle into increasingly smaller segments, he demonstrated that the area could be accurately determined through the limit of these subdivisions. This approach was part of his broader effort to validate the algorithms found in the Nine Chapters, often by reducing complex figures into simpler, fundamental components. Additionally, Liu Hui refined the calculation of pi by employing polygonal methods, which allowed him to achieve a high degree of precision for his time.13