Area of a circle - History behind the formula

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Area of a circle - History behind the formula INTRODUCTION & PROPORTIONALITY Why Circular Shapes? Circular Building Garden Tower for grain storage The Key Idea: Area ( A ) is Proportional to Square of Circumference ( C ) SQUARE: P=4a⟶A=a^2⟶P^2:A=(4a)^2:a^2=16a^2:a^2=16:1 (Constant for all squares) EQUILATERAL TRIANGLE: P=3a→A=√3/4 a^2→P^2:A=(3a)^2:√3/4=9a^2:√3/4=36:√3 (Constant for all equilateral triangles) CIRCLE: C^2: A must be constant! What is it? ANCIENT APPROXIMATIONS & RATIOSBABYLONIANS (Before 1500 BCE) Found C^2:A≈12 A≈C^2/12←ANCIENT GREEKS Knew A/r^2 is constant Approximated it as 256/81 ANCIENT EGYPTIANS ( ∼1500 BCE) More Accurate A≈(8d/9)^2=64/81 d^2=64/81(2r)^2=256/81 r^2Effective π≈3.16 BAUDHĀYANA ŚULBASŨTRA (c. 800 BCE) Amazing, same formula found via geometric methods! A≈256/81 r^2 ARCHIMEDES & THE EXACT FORMULA ARCHIMEDES' BREAKTHROUGH (c. 250 BCE) The Constant is Exactly π ! A=πr^2Regular Polygon Property: (Based on <IMAGE1, Fig 6.36) Triangle Pentagon Hexagon Area =1/2× Perimeter (P)× radius (r) Thought Experiment: more and more So Area " →1/2×C×r=1/2×(2πr)×r=πr^2 sidesNILAKANTHA SOMAYAJI'S VISUAL PROOF THE MOST VISUAL EXPLANATION (c. 1500) STEP 1: Divide the Circle Small slice z line, height = r STEP 2: Rearrange the Slices STEP 3: Analyze the Shape Parallelogram STEP 4: Final Calculation Area of Circle ≈ Area of Parallelogram Area of Circle ± Area of Parallelogram A=" Base × Height A=(πr)×r=πr^2