Chapter 6 — Measuring Space: Perimeter and Area

Chapter 6 of the Class 9 Maths NCERT textbook, "Measuring Space: Perimeter and Area", teaches how to measure the perimeter and area of shapes, covering a circle's circumference (2*pi*r), arc length, sectors, the area of triangles, Heron's formula, and the area of a circle (pi*r^2).

• From Perimeter to Area — The chapter builds from perimeter, the total length around a shape's border, toward area, developing measurement of both for polygons and curved figures like the circle.
• The Constant Pi — Every circle's circumference-to-diameter ratio is the irrational constant pi (about 22/7 or 3.14). Its history runs from ancient approximations to Madhava's infinite series.
• Circle Measurements — Using central angles, the chapter derives arc length and sector area, and proves the area of a circle is pi*r^2, giving formulas for the whole circle and its parts.
• Areas of Polygons — Area formulas for rectangles, parallelograms, and triangles lead to Heron's formula for any triangle from its sides, and Brahmagupta's formula generalises it to cyclic quadrilaterals.

1. 01Perimeter of a circle (circumference) is C = 2*pi*r, with diameter d giving pi*d
2. 02pi is the irrational circumference-to-diameter (C/D) ratio, about 22/7 or 3.14
3. 03Arc length subtending angle t degrees is 2*pi*r * (t/360)
4. 04Area of a triangle equals half base times height (1/2 * b * h)
5. 05Heron's formula: area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
6. 06Area of a circle is A = pi*r^2; area of a sector is pi*r^2 * (t/360)
7. 07Brahmagupta's formula gives the area of a cyclic quadrilateral and generalises Heron's formula