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Mensuration — Measuring the World Around You

Learning Goals

By the end of this chapter, you will be able to:

• Compute areas and perimeters of all standard 2D figures (triangles, quadrilaterals, circles, polygons) without hesitation.
• Master 3D mensuration: surface areas and volumes of prisms, cylinders, cones, spheres, pyramids and frustums.
• Handle combination solids, hollow objects, and "melted-and-recast" problems.
• Apply shortcut ratios (scaling laws) that solve 30-second questions in 10 seconds.
• Spot the classic traps examiners set in SSC CGL Tier-II, IBPS PO Mains, CAT and CSAT.

The Hook: A ₹40 Lakh Mistake

In 2019, a contractor in Pune was hired to paint a cylindrical water tank. He measured the tank's diameter as 4 m and height as 6 m, and calculated the area to be painted as π × 4 × 6 = 24π ≈ 75.4 m². He quoted his price accordingly.

He lost money on the job. Why?

Because the curved surface area of a cylinder is 2πrh — and r is the *radius*, not the diameter. The true area was 2 × π × 2 × 6 = 24π… wait, that matches. His real mistake? He forgot the top circular lid (πr²) and, worse, quoted for the *inside* painting too, which needed the inner radius. One chapter of mensuration, ignored, cost him lakhs.

Mensuration is not "formula rattna." It is the mathematics of *physical space* — and examiners love it precisely because one wrong letter (r vs d, CSA vs TSA) destroys the whole answer.

"But I Keep Forgetting Formulas…"

Every aspirant says this. Here is the truth: you forget formulas because you memorise them as *strings of symbols*. Your brain stores *stories and shapes*, not strings.

So in this chapter, every formula comes attached to a picture in words. Learn the picture, and the formula rebuilds itself in the exam hall — even under stress.

🧠 Active Recall: Before reading further — write down, from memory, the area of a trapezium and the volume of a cone. Check yourself in 2 minutes.

Part 1: The 2D World — Areas and Perimeters

Triangles: The Mother of All Shapes

Every polygon can be cut into triangles. So triangle formulas are the foundation.

• Basic: Area = ½ × base × height
• Heron's formula (when all three sides a, b, c are known):
• s = (a + b + c)/2 (semi-perimeter)
• Area = √[s(s−a)(s−b)(s−c)]
• Equilateral triangle (side a): Area = (√3/4) a²; Height = (√3/2) a
• Right triangle: Area = ½ × (product of the two legs)

Advanced nugget: For any triangle, Area = ½ ab·sinC (two sides and the included angle). CSAT and CAT love this when the height is not given.

Quadrilaterals

• Rectangle: Area = l × b; Perimeter = 2(l + b); Diagonal = √(l² + b²)
• Square: Area = a²; Diagonal = a√2 (so Area = d²/2 — a favourite shortcut!)
• Parallelogram: Area = base × height (the *perpendicular* height, not the slant side — classic trap)
• Rhombus: Area = ½ × d₁ × d₂ (product of diagonals, halved)
• Trapezium: Area = ½ × (sum of parallel sides) × height

🧠 Active Recall: A rhombus and a square both have all sides equal. Why does the rhombus have a *smaller* area for the same side length? (Hint: think of the square being "pushed over" — what shrinks?)

Circles

• Area = πr²; Circumference = 2πr
• Sector (angle θ in degrees): Area = (θ/360) × πr²; Arc length = (θ/360) × 2πr
• Segment: Sector area − triangle area

Memory trick — "Pie Are Square": People say "pie are round," but mathematically, π r² — "pie are squared." You will never swap Area and Circumference again.

Part 2: The Scaling Laws — Your Secret Weapon

This is where Advanced-level thinking begins.

If all linear dimensions of a figure are multiplied by k:

• Perimeter scales by k
• Area scales by k²
• Volume scales by k³

Example: Radius of a sphere is doubled (k = 2). Surface area becomes 4×, volume becomes 8×. No calculation needed — this single idea solves dozens of exam questions.

🧠 Active Recall: The side of a cube is increased by 50% (k = 1.5). By what percent does its volume increase? (Answer: 1.5³ = 3.375 → 237.5% increase. Most students wrongly say 150%.)

Part 3: The 3D World — Surface Area and Volume

The Big Family Tree

Prism family (uniform cross-section): Volume = (base area) × height. That's it. Cube, cuboid, cylinder — all are prisms.

• Cuboid: V = lbh; TSA = 2(lb + bh + hl); Diagonal = √(l² + b² + h²)
• Cube: V = a³; TSA = 6a²; Diagonal = a√3
• Cylinder: V = πr²h; CSA = 2πrh; TSA = 2πr(h + r)

Pointed family (comes to an apex): Volume = ⅓ × (base area) × height. The "one-third" is the soul of this family.

• Cone: V = ⅓πr²h; CSA = πrl where l = √(r² + h²); TSA = πr(l + r)
• Pyramid: V = ⅓ × base area × h

Sphere family:

• Sphere: V = (4/3)πr³; Surface area = 4πr²
• Hemisphere: V = (2/3)πr³; CSA = 2πr²; TSA = 3πr² (curved + flat circle)

🧠 Active Recall: Why is a hemisphere's TSA = 3πr² and not 2πr²? Say the reason out loud before checking above.

The Frustum — The Examiner's Darling

Cut the top off a cone parallel to the base: you get a frustum (think of a bucket).

• V = (⅓)πh(R² + r² + Rr)
• CSA = πl(R + r), where l = √(h² + (R−r)²)

Buckets, glasses, and lampshades in exams are always frustums.

Part 4: The Three Classic Problem Types

Type 1: Melted and Recast

"A metallic sphere of radius 9 cm is melted and recast into small spheres of radius 3 cm. How many?"

Golden rule: Volume is conserved. Number = Big volume / Small volume = (9/3)³ = 27. (Notice — the scaling law again!)

Type 2: Surface Area Changes on Joining/Cutting

When two cubes are joined, two faces disappear from the exposed surface. When a cube is cut, new faces appear. Always track *which faces* are lost or gained — never recompute from scratch.

Type 3: Cost, Painting, Fencing, Carpeting

These are mensuration wearing a disguise. Convert words to geometry:

• "Carpet the floor" → area of rectangle
• "Paint the walls" → area of 4 walls = 2h(l + b) — no floor, no ceiling!
• "Fence the field" → perimeter
• "Plaster inside a tank open at top" → CSA + base, not TSA

🧠 Active Recall: A room is painted on walls AND ceiling. Write the expression for the painted area. (Answer: 2h(l+b) + lb.)

Mnemonic: "CONE has ONE-third, SPHERE has FOUR-thirds"

• CONE → the word has "ONE" hidden in it → ⅓ πr²h
• SPHERE → "FOUR" rhymes with the round "O" of a ball → 4/3 πr³ and 4πr²

One glance at the word, and the fraction appears. Try forgetting it now — you can't.

Summary Table

Scaling law: length ×k → area ×k² → volume ×k³.
Conservation law: melting/recasting keeps volume constant.

🔁 Review this chapter after 2 days.

Close the book and rebuild every formula from the pictures: the "one-third" pointed family, "pie are squared," and the k–k²–k³ ladder. If you can rebuild them, you own them.