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"Measuring Space: Perimeter and Area" is Chapter 6 of Ganita Manjari Part I, the new NCERT Class 9 maths book for the 2026-27 session (first edition, April 2026). Alongside familiar formulas it adds the history of π, arc length, Brahmagupta's formula and an old Indian construction.
What does Chapter 6 of Ganita Manjari cover?
The chapter PDF opens with lane staggers on a running track, then covers:
Perimeter as the total length around a shape, and the C/D ratio: circumference divided by diameter is the same for every circle.
Estimates of π from Archimedes, Zu Chongzhi and Āryabhaṭa to Mādhava's exact infinite series, and why π is irrational.
Arc length, then the areas of rectangles, parallelograms and triangles, including the fact that a median splits a triangle into two triangles of equal area.
Heron's formula, and Brahmagupta's formula for a cyclic quadrilateral (the book calls it a cyclic 4-gon).
Baudhāyana's method of squaring a rectangle, the area of a circle, and sectors and segments.
The book's own summary lists the key results: circumference 2πr, arc length 2πr × θ/360°, area of a circle πr², and sector area πr² × θ/360° (NCERT chapter 6).
How do you find arc length and the area of a sector?
Take a sector of radius 6 cm with a central angle of 120°, as in the diagram above. 120° is one third of 360°, so every answer is one third of the full-circle value.
Arc length = 2πr × 120/360 = 2π × 6 ÷ 3 = 4π cm ≈ 12.57 cm.
Perimeter of the sector = arc + two radii = 12.57 + 6 + 6 = 24.57 cm.
Area of the sector = πr² × 120/360 = π × 36 ÷ 3 = 12π cm² ≈ 37.70 cm².
Forgetting the two radii in step 2 is a common slip; our post on the perimeter of a sector has more practice sums on it.
How do you use Heron's formula step by step?
Heron's formula gives a triangle's area from its three sides alone: if the sides are a, b and c and s is the semi-perimeter (a + b + c)/2, the area is √[s(s − a)(s − b)(s − c)] (NCERT chapter 6).
Try a triangle with sides 9 cm, 10 cm and 17 cm (the left shape in the diagram).
Check it is a triangle: 9 + 10 = 19, which is more than 17.
s = (9 + 10 + 17) ÷ 2 = 18 cm.
s − a = 9, s − b = 8, s − c = 1.
Area = √(18 × 9 × 8 × 1) = √1296 = 36 cm².
Check it another way. Area = ½ × base × height, so the height on the 17 cm side is 2 × 36 ÷ 17 = 72/17 ≈ 4.24 cm. Its foot splits that side into 154/17 cm and 135/17 cm, and Pythagoras agrees: (154/17)² + (72/17)² = 100 = 10² and (135/17)² + (72/17)² = 81 = 9². For more worked examples, see our guide to Heron's formula in Class 9.
A quick check with your own numbers: a 3 cm by 4 cm rectangle is cyclic, s = 7, and √(4 × 3 × 4 × 3) = 12 cm², which matches length × width.
What mistakes cost marks in this chapter?
Radius or diameter? Circumference is πd or 2πr. Halve the diameter before using any r formula.
Half the perimeter, not all of it. In Heron's formula s is (a + b + c)/2.
Wrong height. In ½ × base × height, the height must be perpendicular to the base you chose; our guide to which height goes with which base shows why.
Writing π = 22/7. The book stresses π ≈ 22/7, never π = 22/7. Use the value the question gives.
Square units. Area answers need cm² or m²; perimeter answers do not.
If you keep choosing the wrong method rather than slipping on arithmetic, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
How many marks is mensuration in CBSE Class 9 2026-27?
In the CBSE Class IX Mathematics curriculum for 2026-27, Unit V Mensuration carries 14 of the 80 theory marks, and that unit includes Surface Area and Volume as well as Area and Perimeter. Internal assessment is a further 20 marks. Its learning outcomes include using Heron's formula, areas of sectors and segments, and explaining why Heron's formula is a special case of Brahmagupta's.
A simple plan for the week before a test:
Write the summary formulas from memory, then check them against the book.
Do one sector, one Heron and one circle-area sum with your own numbers, checking each a second way.
Explain the median result and the d = 0 link aloud, in two sentences each.
