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Archimedes' Method of Exhaustion Archimedes' Method of Exhaustion Archimedes calculated π by finding bounds using polygons: Inscribed Polygons Always smaller than the circle Provides a lower bound for π Circumscribed Polygons Always larger than the circle Provides an upper bound for π As the number of sides increases, the bounds get tighter. Using 96-sided polygons, Archimedes proved: 3.1408 < π < 3.1429 Liu Hui's Method (China, 263 CE) Liu Hui's Method (China, 263 CE) Liu Hui refined the calculation of π using many-sided polygons: Starting with a 6-sided polygon Then 12 sides Then 24 sides Doubling to 48, 96, 192, 384... Liu Hui used the principle of repeatedly doubling the number of sides to achieve greater precision. With a 3072-sided polygon, he calculated: π ≈ 3.14159 The Magic Circle Number! 1 Diameter = 1 Circumference = 3.14 3.14 ÷ 1 = 3.14 2 Diameter = 2 Circumference = 6.28 6.28 ÷ 2 = 3.14 3 Diameter = 3 Circumference = 9.42 9.42 ÷ 3 = 3.14 4 Diameter = 4 Circumference = 12.56 12.56 ÷ 4 = 3.14 No matter what size cookie: Circumference ÷ Diameter = 3.14159... = π This special number π (Pi) never changes!