GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6
Study Materials
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TRACE — Understanding Perimeter
Perimeter is the total length of a closed boundary, counted once. It is a length, measured in units such as cm or m.
Identify the actual boundary. List its lengths in one unit. Add them once, and check that you returned to your starting point.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.1 · Figures 6.2–6.4 · printed pages 119.
Worked example →DISCOVER — Circumference and π
All circles share the ratio = π. Diameter is twice radius. Physical measurements estimate this ratio; they do not give its exact value.
Identify radius or diameter. Convert d = 2r if needed. Keep π in an exact answer unless an approximation is requested.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.2 · Figures 6.5–6.7 · printed pages 120–122.
Worked example →APPROXIMATE — Understanding π
π cannot be written as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating. This is a theorem, not a conclusion proved by inspecting a few digits.
Separate an exact symbol from an approximation. Use the value and rounding requested by the problem. Retain extra working digits before rounding the final result.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.3 · π Is Irrational · printed pages 123–125.
Worked example →ARC — Arc Length and Athletics Track
An arc subtending θ degrees has θ/360 of the full circumference. On a stadium-shaped track the extra distance in outer lanes comes from the bends.
Use the central angle. Find its fraction of 360°. Multiply by circumference. Add straight edges only when a whole boundary is requested.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.4 · Figures 6.8–6.11 · printed pages 125–127.
Worked example →PUZZLE — Perimeter Reasoning
Semicircular arc length is proportional to diameter. Interior dotted arcs are not part of the outer boundary of a union of circles.
Mark the boundary being measured. Express each arc as a fraction of its circumference, then use relationships between diameters or central angles.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.5 · Examples 1–2, Figures 6.12–6.13 · printed pages 127–128.
Worked example →RESHAPE — Rectangle and Parallelogram Area
Rectangle area is ab. A parallelogram can be cut and rearranged to a rectangle with the same base and perpendicular height; its area is bh.
Choose a base, identify its perpendicular height, then multiply. For a thin parallelogram, repeated finite cuts can move the overhang before forming a rectangle.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.6–6.7 · Figures 6.16–6.19 · printed pages 130–132.
Worked example →TRIANGLE — Discovering Triangle Area
Two copies form a parallelogram, so triangle area is half base times perpendicular height. A median splits a triangle into equal areas because the two bases are equal and share an altitude.
Choose a base and its perpendicular height (possibly outside the triangle). A median joins a vertex to the midpoint of the opposite side.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.8 · Figures 6.20–6.22 · printed pages 132–134.
Worked example →SIDES — Heron, Brahmagupta and Generalisation
Heron’s formula applies to valid triangles. Brahmagupta’s formula applies to cyclic quadrilaterals. A general quadrilateral’s sides alone do not determine its area.
Check triangle inequalities first. Compute semiperimeter, substitute, and retain the radical exactly when possible. For four sides, first establish that the quadrilateral is cyclic.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.8.1 · Examples 3–7, Figures 6.23–6.29 · printed pages 134–140.
Worked example →TRANSFORM — Squaring a Rectangle
Baudhāyana’s construction uses a right triangle and the difference of two squares to construct a square of area ab.
For a > b > 0, follow E, F, the auxiliary square AFGH, arc intersection K, perpendicular KP, and the square HPQS. Verify the length using the right triangle HKP.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.9 · Figure 6.30 and starred proof · printed pages 140–142.
Worked example →CIRCLE — Area, Sector and Applications
Rearranged sectors approach a parallelogram of base πr and height r. Thus circle area is π . A sector uses two radii; a segment uses a chord and an arc.
For a sector use the central-angle fraction of circle area. For a minor segment with angle at most 180°, subtract the triangle between the radii from the sector.
Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.10–6.10.1 · Figures 6.36–6.40 · printed pages 144–148.
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