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Starting with a simple hexagon.

Liu Hui (3rd Century)

Liu Hui, a great mathematician of the state of Cao Wei, provided a detailed commentary on the Nine Chapters on the Mathematical Art.

Using an inscribed polygon, he successfully calculated the area with a 96-sided polygon to demonstrate that Pi is approximately 3.14 or 157/50.

He noted that the larger the number of sides, the closer the area becomes to the circle, leaving no area left out.

Zu Chongzhi (5th Century)

Zu Chongzhi extended Liu Hui's method. He proved mathematically that Pi lies strictly between 3.1415926 and 3.1415927.

He provided two rational approximations for Pi:

  • Accurate: 22 / 7
  • Highly Accurate: 355 / 113 (Milü)
Zu Chongzhi's original text is lost. Modern accounts associate his result with 12,288 sides.

Mathematical Data

Sides (n): 6
Radius (r): 1.00000
Length (s): 1.00000
Perim (n×s): 6.00000000
Est. π: 3.00000000
True π: 3.14159265...
Abs Error: 0.14159265

Key Milestones

6 sides (Hexagon)
96 sides (~3.14)
3072 sides (~3.14159)
12288 sides (355/113)