Mensuration 2D & 3D Short Notes | Quick Revision for Exams

MENSURATION (2D & 3D) – COMPLETE SHORT NOTES FOR QUICK REVISION

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PART A — 2D MENSURATION

1. Square

• Side = a

• Area = a²

• Perimeter = 4a

• Diagonal = a√2

• When diagonal = d → a = d/√2

• % change rule:

  • If side ↑ x%, area ↑ (x + x²/100)%

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2. Rectangle

• Length = L, Breadth = B

• Area = L × B

• Perimeter = 2(L + B)

• Diagonal = √(L² + B²)

• If perimeter fixed → area max when L = B (i.e., a square)

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3. Parallelogram

• Area = base × height

• Area = a × b × sinθ

• Perimeter = 2(a + b)

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4. Triangle

General

• Area = ½ × base × height

• Heron's Formula:

  • s = (a + b + c)/2

  • Area = √(s(s – a)(s – b)(s – c))

Special Triangles

• Equilateral Triangle:

  • Side = a

  • Area = (√3/4)a²

  • Height = (√3/2)a

  • Perimeter = 3a

• Right Triangle:

  • Area = ½ × product of perpendicular sides

  • Hypotenuse² = a² + b²

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5. Trapezium

• Parallel sides = a, b ; Height = h

• Area = ½ × (a + b) × h

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6. Circle

• Radius = r, Diameter = 2r

• Area = πr²

• Circumference = 2πr

• Arc length = (θ/360) × 2πr

• Sector area = (θ/360) × πr²

• Chord length = 2r sin(θ/2)

Important Pi Values

• π = 22/7 (approx)

• π ≈ 3.14

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7. Semi-circle

• Area = ½ πr²

• Perimeter = πr + 2r

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8. Rhombus

• Area = ½ × d₁ × d₂
  (d₁, d₂ are diagonals)

• Side = √[(d₁² + d₂²)/4]

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9. Polygon (Regular)

• Sum of interior angles = (n – 2) × 180°

• Interior angle = [(n – 2)/n] × 180°

• Exterior angle = 360°/n

Area of Regular Polygon

• Area = (n × a²) / (4 tan(π/n))
  (a = side)

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PART B — 3D MENSURATION

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1. Cube

• Side = a

• TSA (Total Surface Area) = 6a²

• LSA (Lateral Surface Area) = 4a²

• Volume = a³

• Diagonal of cube = a√3

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2. Cuboid

• Sides = L, B, H

• TSA = 2(LB + BH + HL)

• LSA = 2H(L + B)

• Volume = L × B × H

• Diagonal = √(L² + B² + H²)

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3. Cylinder

• Radius = r, Height = h

Formulas

• TSA = 2πr(h + r)

• CSA = 2πrh

• Volume = πr²h

• Curved area ∝ radius × height

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4. Cone

• Radius = r, Height = h, Slant height l = √(r² + h²)

Formulas

• CSA = πrl

• TSA = πr(r + l)

• Volume = (1/3)πr²h

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5. Sphere

• Radius = r

Formulas

• TSA = 4πr²

• Volume = (4/3)πr³

• Circle → 2D; Sphere → 3D

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6. Hemisphere

• Curved Area = 2πr²

• TSA = 3πr²

• Volume = (2/3)πr³

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7. Frustum of Cone

If a cone is cut parallel to its base:

• Top radius = r₁

• Bottom radius = r₂

• Height = h

• Slant height = l = √[(r₁ – r₂)² + h²]

Formulas

• CSA = π (r₁ + r₂) l

• TSA = π (r₁ + r₂) l + π(r₁² + r₂²)

• Volume = (1/3)πh (r₁² + r₂² + r₁r₂)

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PART C — IMPORTANT SHORTCUTS & ONE-LINERS

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1. Area increases with the square of scale factor

• If dimensions increase by x%
  → Area increases by (x + x²/100)%
  → Volume increases by (3x + 3x²/100 + x³/10000)%

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2. Largest area/perimeter results

• Among all rectangles with fixed perimeter → Square has max area

• Among all closed curves → Circle has max area

• Among shapes with fixed surface area → Sphere has max volume

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3. Ration of sides to areas

• If side ratio = a : b
  → Area ratio = a² : b²
  → Volume ratio = a³ : b³

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4. Units Conversion

• 1 m = 100 cm

• 1 m² = 10,000 cm²

• 1 m³ = 10,00,000 cm³ = 1000 L

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5. Path Problems

• Path outside square:

  • Area = (outer side)² – (inner side)²

• Path inside rectangle:

  • Area = LB – (L – 2W)(B – 2W)

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6. Shaded region type questions

Use formulas:

• Shaded = Total area – unshaded

• For quadrant, semicircle, concentric circles → apply πr² parts.

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PART D — HIGH-SCORING PRACTICE CONCEPTS

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1. Paper folding (Mirror images)

• When shape is folded → perimeter changes

• When unfolded → area remains same

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2. Increase in diagonal increases area?

• Increases only if aspect ratio stays same.

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3. Mensuration Mixture (Cube inside sphere etc.)

Common examples:

• Cube inside sphere:

  • diagonal = diameter → a√3 = 2r

• Sphere inside cube:

  • cube side = 2r