The fraction of the rectangle covered by n identical tangent circles in a row is always constant at π / 4 approximately 11/14 (or 78.5%) regardless of whether n equals 10, 20 or 50.
Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!
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Conjecture: The fraction of the rectangle covered by any number n of identical circles packed side-by-side in a row is independent of n and always equals π / 4.
Test particular cases:
For 10 circles: Area of circles / Area of rectangle = [10 x (π/4)d²] / [10d x d] = π / 4 approximately 11/14.
For 20 circles: [20 x (π/4)d²] / [20d²] = π / 4 approximately 11/14.
For 50 circles: [50 x (π/4)d²] / [50d²] = π / 4 approximately 11/14.
Proof:
Let each circle have diameter d and radius r = d/2.
A row of n circles forms a rectangle of width nd and height d, giving area nd².
Total area of n circles is n x π(d/2)² = n x (π/4)d².
The ratio is [n(π/4)d²] / [nd²] = π / 4, which is constant for all n.
For more NCERT Solutions of Class 9 Ganita Manjari Chapter 6 Measuring Space: Perimeter and Area Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-6/