Chapter 6: Measuring Space: Perimeter and AreaClass 9 Mathematics — summary, notes, extra questions & NCERT solutions

Summary

This chapter measures the boundary and the region of plane shapes. The perimeter is the total distance around a shape: \(4a\) for a square of side \(a\), and \(2(l+w)\) for a rectangle. For a circle the perimeter is the circumference \(C = 2\pi r = \pi d\), where \(\pi\) is the constant ratio of circumference to diameter; historical estimates include Archimedes' bound \(3\tfrac{10}{71} < \pi < 3\tfrac{1}{7}\), Aryabhata's \(\pi \approx 3.1416\) and Madhava's exact infinite series. Area measures the surface enclosed: a rectangle has area \(l \times w\) and a triangle \(\dfrac{1}{2} \times \text{base} \times \text{height}\). When the height is not known, Heron's formula gives the area of a triangle from its sides: \(\sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \dfrac{a+b+c}{2}\) is the semi-perimeter. The area of a circle is \(\pi r^2\), explained by Nilakantha as equal to a triangle with base \(C\) and height \(r\). The chapter also treats the area of a sector as a fraction of the whole circle and applies these ideas to real problems such as relay-race lane staggers.

Key terms

Extra questions & answers

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HOME MEASUREMENT

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Exercise Set 6.1

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Exercise Set 6.2

Exercise Set 6.3

End-of-Chapter Exercises

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