Measuring Space | Class 9 Mathematics Notes

6.2 PERIMETER OF A CIRCLE—THE C/D RATIO

This section explores the perimeter of a circle, called the circumference, and the constant ratio of circumference to diameter, known as π (pi). Ancient civilizations observed that this ratio remains constant regardless of the size of the circle. The section describes a simple home experiment to estimate π by measuring the diameter of a cotton reel and the length of thread wrapped around it multiple times, calculating the ratio L/(20D). Historical approximations of π are discussed, including the Mesopotamian value 3.125, Archimedes' method of inscribed and circumscribed polygons bounding π between 3 and approximately 3.46, and later refinements by Ptolemy, Liu Hui, Zu Chongzhi, Aryabhata, and Brahmagupta. The section highlights the infinite series formula for π discovered by Mādhava, marking the beginning of calculus. It concludes by noting that π is irrational, meaning it cannot be expressed exactly as a fraction, and its decimal expansion is infinite and non-repeating.

📊 Diagram: Fig. 6.6 The Mesopotamian Hexagon-to-Circle comparison; Fig. 6.7: Archimedes’ method utilising inscribed and circumscribed polygons.

🧪 Activity: Home Measurement: Measure diameter and thread length to estimate π; reflect on historical approximations.

🔗 Connection: Prepares for the next section on calculating lengths of arcs of circles using π.