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SSC-Standard Mathematics for Forest Posts · Chapter 7

Mensuration: 2D and 3D

What to remember

  • Mensuration measures lengths, areas and volumes. Area is in square units, volume in cubic units, and 1 m³ = 1000 litres. Use π = 22/7 unless told otherwise.
  • Learn the 2D formulas for the triangle, rectangle, parallelogram, rhombus, trapezium, circle and sector, and the 3D formulas for the cube, cuboid, cylinder, cone, sphere and hemisphere.
  • When a solid is melted and recast, the volume stays the same. Use volume equality to count the new pieces.

Plane figures: area and perimeter

FigureAreaPerimeter
Rectangle (l, b)l × b2(l + b)
Square (a)a² = d²/2 (d = diagonal)4a
Triangle½ × base × heightsum of sides
Equilateral triangle (a)(√3/4) a²3a
Heron's formula (sides a, b, c, s = half the perimeter)√[s(s − a)(s − b)(s − c)]a + b + c
Parallelogrambase × height2(sum of adjacent sides)
Rhombus½ × d₁ × d₂4 × side
Trapezium½ × (sum of parallel sides) × heightsum of sides

Worked examples.

  • A rectangle 12 × 5 has area 60. A square of area 144 has side 12 and perimeter 48.
  • A triangle with base 10 and height 7 has area 35.
  • Sides 13, 14, 15: s = 21, area = √(21 × 8 × 7 × 6) = √7056 = 84.
  • An equilateral triangle of side 8 has area (√3/4) × 64 = 16√3.
  • A rhombus with diagonals 10 and 24 has area 120. A trapezium with parallel sides 9 and 15 and height 8 has area 96.
  • A parallelogram with base 12 and height 7 has area 84.
  • A rectangle with perimeter 30 and length 9 has breadth 6 and area 54. A square with diagonal 12 has area 72.
  • A park 20 × 15 with a 2 m path all round outside: outer rectangle 24 × 19 = 456, so path area = 456 − 300 = 156.

Circle, sector and ring

QuantityFormula (radius r, angle θ at centre)
Circumference2πr
Areaπr²
Arc length(θ/360°) × 2πr
Area of sector(θ/360°) × πr²
Area of ring (outer R, inner r)π(R² − r²)
Perimeter of semicircleπr + 2r

Worked examples.

  • r = 7: area = 22/7 × 49 = 154. r = 14: circumference = 2 × 22/7 × 14 = 88.
  • Sector of angle 60° and r = 21: area = 1/6 × 22/7 × 441 = 231. Arc length = 1/6 × 2 × 22/7 × 21 = 22.
  • Ring with R = 14 and r = 7: area = 22/7 × (196 − 49) = 462.
  • A wire of length 44 bent into a circle has 2πr = 44, so r = 7 and area 154.
  • A semicircle of radius 7 has perimeter 22 + 14 = 36.

Scaling. If the radius of a circle increases by 10%, the area becomes (1.1)² = 1.21 times, an increase of 21%.

Solids: surface area and volume

SolidVolumeCurved surface area (CSA)Total surface area (TSA)
Cube (a)a³4a² (four walls)6a²
Cuboid (l, b, h)lbh2h(l + b)2(lb + bh + hl)
Cylinder (r, h)πr²h2πrh2πr(r + h)
Cone (r, h, slant l)⅓πr²hπrlπr(r + l)
Sphere (r)(4/3)πr³4πr²4πr²
Hemisphere (r)(2/3)πr³2πr²3πr²
Prismbase area × heightperimeter of base × heightCSA + 2 × base area

Other relations:

  • Cone: l² = r² + h². Cube diagonal = a√3. Cuboid diagonal = √(l² + b² + h²).
  • Frustum of a cone (radii R and r, height h): volume = (πh/3)(R² + Rr + r²).

Worked examples.

  • Cube a = 5: volume 125, TSA 150, diagonal 5√3.
  • Cuboid 8 × 6 × 5: volume 240, TSA 2(48 + 30 + 40) = 236. A cuboid 4 × 3 × 12 has diagonal √(16 + 9 + 144) = 13.
  • Cylinder r = 7, h = 10: volume = 22/7 × 49 × 10 = 1540, CSA = 2 × 22/7 × 7 × 10 = 440.
  • Cone r = 7, h = 24: slant = √(49 + 576) = 25, volume = ⅓ × 22/7 × 49 × 24 = 1232, CSA = 22/7 × 7 × 25 = 550.
  • Sphere r = 7: surface area = 4 × 22/7 × 49 = 616. Sphere r = 21: volume = 4/3 × 22/7 × 9261 = 38808.
  • Hemisphere r = 7: CSA = 308, TSA = 462.
  • A triangular prism with base 6, height 4 and length 10: volume = 12 × 10 = 120.
  • A tank 5 m × 4 m × 3 m holds 60 m³ = 60,000 litres.

Change of shape and ratios

Melting and recasting. Volume before = volume after. A sphere of radius 6 recast into small spheres of radius 2 gives (6/2)³ = 27 spheres. A wire drawn from a metal block has volume πr²h.

Scaling rules.

  • If every linear dimension is multiplied by k, area becomes k² times and volume k³ times. Doubling a cube's edge makes the surface area 4 times and the volume 8 times.
  • Cylinders of equal height with radii in the ratio 2 : 3 have volumes in the ratio 4 : 9.
  • A cone and a cylinder with the same base and height have volumes in the ratio 1 : 3.

Units. 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 10⁶ cm³. 1 litre = 1000 cm³ and 1 m³ = 1000 litres.

For forestry-type problems, treat a log as a cylinder and a stack of wood as a cuboid. Use cross-section area × length for volume.

Practical mensuration for forestry-type problems

  • Log or pole. A log is a cylinder. A log of radius 0.35 m and length 10 m has volume (22/7) × 0.35 × 0.35 × 10 = 3.85 m³. Always use the same unit for radius and length.
  • Timber stack. A rectangular stack of wood is a cuboid. Its volume is length × breadth × height. A stack 6 m × 2 m × 1.5 m holds 18 m³.
  • Water tank or trench. A trench is a cuboid, a circular well is a cylinder. Digging a well of radius r and depth h removes πr²h of soil, which spread over a rectangular field raises it by (volume) / (area of field).
  • Tree trunk cone. The cone formula is used for tent-shaped heaps. A heap of grain in the form of a cone with radius 7 m and height 3 m has volume ⅓ × 22/7 × 49 × 3 = 154 m³.
  • Fencing and planting. The perimeter gives the length of fence needed. The area divided by the area per plant gives the number of plants. A rectangular plot 50 m × 30 m with one plant per 5 m² has 1500 / 5 = 300 plants.
  • Combination of solids. A solid made of a cylinder with a hemisphere on top has volume πr²h + (2/3)πr³ and surface area 2πrh + 2πr² + πr² (curved surface of the cylinder, hemisphere and the base).

Conversions and common slips

QuantityConversion
1 hectare10,000 m²
1 km²100 hectares
1 m²10,000 cm²
1 litre1000 cm³
1 m³1000 litres

Always check that all lengths use the same unit before substituting into a formula. For a closed cylinder the total surface area is the curved surface plus the two circular ends, 2πr(r + h).

Exam traps

  • Using the radius when the diameter is given, or the other way round.
  • Using the height h instead of the slant height l in the CSA of a cone.
  • Forgetting the ⅓ in the volume of a cone, or the 4/3 in the volume of a sphere.
  • Confusing CSA with TSA. TSA includes the circular ends (and for a hemisphere, the flat top).
  • Using the slant side instead of the perpendicular height in the area of a parallelogram or trapezium.
  • Leaving out the square when the radius changes by a percentage (area changes as r²).
  • Forgetting to convert units (cm to m, or m³ to litres) before the final answer.
  • Applying k² for volume. Volume scales as k³; only area scales as k².

One-liners

  • 1. Area of a triangle = ½ × base × height.
  • 2. Heron's formula: area = √[s(s − a)(s − b)(s − c)].
  • 3. Area of an equilateral triangle = (√3/4) a².
  • 4. Area of a rhombus = ½ × d₁ × d₂.
  • 5. Area of a trapezium = ½ × (a + b) × h.
  • 6. Area of a circle = πr²; circumference = 2πr.
  • 7. Volume of a cylinder = πr²h; CSA = 2πrh.
  • 8. Volume of a cone = ⅓πr²h; slant height l = √(r² + h²).
  • 9. Volume of a sphere = (4/3)πr³; surface area = 4πr².
  • 10. TSA of a hemisphere = 3πr².
  • 11. Volume of a cube = a³; TSA = 6a².
  • 12. 1 m³ = 1000 litres.

Practice questions

  1. The area of a triangle with base 10 cm and height 7 cm is

    1. 24.5 cm²
    2. 17 cm²
    3. 35 cm²
    4. 70 cm²
    Answer

    C. 35 cm²

    ½ × 10 × 7 = 35.

  2. The perimeter of a square whose area is 144 cm² is

    1. 36 cm
    2. 12 cm
    3. 144 cm
    4. 48 cm
    Answer

    D. 48 cm

    Side = 12, perimeter = 4 × 12 = 48.

  3. The area of a triangle with sides 13 cm, 14 cm and 15 cm is

    1. 91 cm²
    2. 42 cm²
    3. 105 cm²
    4. 84 cm²
    Answer

    D. 84 cm²

    s = 21; √(21 × 8 × 7 × 6) = 84.

  4. The area of an equilateral triangle of side 8 cm is

    1. 8√3 cm²
    2. 16√3 cm²
    3. 64√3 cm²
    4. 32√3 cm²
    Answer

    B. 16√3 cm²

    (√3/4) × 64 = 16√3.

  5. The area of a circle of radius 7 cm (π = 22/7) is

    1. 154 cm²
    2. 308 cm²
    3. 44 cm²
    4. 49 cm²
    Answer

    A. 154 cm²

    22/7 × 49 = 154.

  6. The circumference of a circle of radius 14 cm (π = 22/7) is

    1. 616 cm
    2. 44 cm
    3. 88 cm
    4. 154 cm
    Answer

    C. 88 cm

    2 × 22/7 × 14 = 88.

  7. The area of a rhombus with diagonals 10 cm and 24 cm is

    1. 120 cm²
    2. 34 cm²
    3. 130 cm²
    4. 240 cm²
    Answer

    A. 120 cm²

    ½ × 10 × 24 = 120.

  8. The area of a trapezium with parallel sides 9 cm and 15 cm and height 8 cm is

    1. 72 cm²
    2. 96 cm²
    3. 192 cm²
    4. 120 cm²
    Answer

    B. 96 cm²

    ½ × (9 + 15) × 8 = 96.

  9. The area of a sector of angle 60° in a circle of radius 21 cm (π = 22/7) is

    1. 1386 cm²
    2. 22 cm²
    3. 462 cm²
    4. 231 cm²
    Answer

    D. 231 cm²

    1/6 × 22/7 × 441 = 231.

  10. The length of an arc of angle 60° in a circle of radius 21 cm (π = 22/7) is

    1. 11 cm
    2. 44 cm
    3. 22 cm
    4. 132 cm
    Answer

    C. 22 cm

    1/6 × 2 × 22/7 × 21 = 22.

  11. The area of a ring with outer radius 14 cm and inner radius 7 cm (π = 22/7) is

    1. 462 cm²
    2. 154 cm²
    3. 308 cm²
    4. 616 cm²
    Answer

    A. 462 cm²

    22/7 × (196 − 49) = 462.

  12. The volume of a cube of edge 5 cm is

    1. 150 cm³
    2. 125 cm³
    3. 75 cm³
    4. 25 cm³
    Answer

    B. 125 cm³

    5³ = 125.

  13. The total surface area of a cube of edge 5 cm is

    1. 125 cm²
    2. 100 cm²
    3. 150 cm²
    4. 30 cm²
    Answer

    C. 150 cm²

    6 × 25 = 150.

  14. The diagonal of a cuboid 4 cm × 3 cm × 12 cm is

    1. 19 cm
    2. 12.5 cm
    3. 5 cm
    4. 13 cm
    Answer

    D. 13 cm

    √(16 + 9 + 144) = √169 = 13.

  15. The volume of a cylinder of radius 7 cm and height 10 cm (π = 22/7) is

    1. 154 cm³
    2. 1540 cm³
    3. 4620 cm³
    4. 440 cm³
    Answer

    B. 1540 cm³

    22/7 × 49 × 10 = 1540.

  16. The curved surface area of a cylinder of radius 7 cm and height 10 cm (π = 22/7) is

    1. 440 cm²
    2. 1540 cm²
    3. 308 cm²
    4. 880 cm²
    Answer

    A. 440 cm²

    2 × 22/7 × 7 × 10 = 440.

  17. A cone has radius 7 cm and height 24 cm. Its slant height is

    1. 23 cm
    2. 17 cm
    3. 31 cm
    4. 25 cm
    Answer

    D. 25 cm

    √(49 + 576) = 25.

  18. The volume of a cone of radius 7 cm and height 24 cm (π = 22/7) is

    1. 616 cm³
    2. 3696 cm³
    3. 1232 cm³
    4. 550 cm³
    Answer

    C. 1232 cm³

    ⅓ × 22/7 × 49 × 24 = 1232.

  19. The curved surface area of a cone of radius 7 cm and slant height 25 cm (π = 22/7) is

    1. 1100 cm²
    2. 1232 cm²
    3. 550 cm²
    4. 154 cm²
    Answer

    C. 550 cm²

    π r l = 22/7 × 7 × 25 = 550.

  20. The surface area of a sphere of radius 7 cm (π = 22/7) is

    1. 1437 cm²
    2. 462 cm²
    3. 154 cm²
    4. 616 cm²
    Answer

    D. 616 cm²

    4 × 22/7 × 49 = 616.

  21. The volume of a sphere of radius 21 cm (π = 22/7) is

    1. 12936 cm³
    2. 38808 cm³
    3. 5544 cm³
    4. 1848 cm³
    Answer

    B. 38808 cm³

    4/3 × 22/7 × 9261 = 38808.

  22. The total surface area of a solid hemisphere of radius 7 cm (π = 22/7) is

    1. 462 cm²
    2. 616 cm²
    3. 308 cm²
    4. 154 cm²
    Answer

    A. 462 cm²

    3πr² = 3 × 154 = 462.

  23. A metal sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. The number of small spheres is

    1. 9
    2. 3
    3. 27
    4. 36
    Answer

    C. 27

    The number is (6/2)³ = 27 because the volume is conserved.

  24. A cone and a cylinder have the same base and height. The ratio of the volume of the cone to that of the cylinder is

    1. 1 : 2
    2. 1 : 3
    3. 2 : 3
    4. 3 : 1
    Answer

    B. 1 : 3

    The cone's volume is one-third of the cylinder's.

  25. If the edge of a cube is doubled, its volume becomes

    1. 8 times
    2. 2 times
    3. 6 times
    4. 4 times
    Answer

    A. 8 times

    Volume scales as k³ = 8.

  26. The radius of a circle is increased by 10%. The area increases by

    1. 10%
    2. 20%
    3. 11%
    4. 21%
    Answer

    D. 21%

    (1.1)² = 1.21.

  27. A rectangular tank is 5 m × 4 m × 3 m. How many litres of water can it hold when full?

    1. 60
    2. 600
    3. 6,000
    4. 60,000
    Answer

    D. 60,000

    Volume 60 m³ and 1 m³ = 1000 litres.

  28. A wire 44 cm long is bent into a circle (π = 22/7). The area enclosed is

    1. 49 cm²
    2. 154 cm²
    3. 616 cm²
    4. 308 cm²
    Answer

    B. 154 cm²

    2πr = 44 gives r = 7 and area 154.

  29. A square has a diagonal of 12 cm. Its area is

    1. 36 cm²
    2. 48 cm²
    3. 72 cm²
    4. 144 cm²
    Answer

    C. 72 cm²

    Area = d²/2 = 72.

  30. A 20 m × 15 m park has a 2 m wide path all round it outside. The area of the path is

    1. 156 m²
    2. 136 m²
    3. 140 m²
    4. 176 m²
    Answer

    A. 156 m²

    Outer 24 × 19 = 456, minus 300 = 156.

  31. A right triangular prism has a triangular base with base 6 cm and height 4 cm, and length 10 cm. Its volume is

    1. 60 cm³
    2. 24 cm³
    3. 120 cm³
    4. 240 cm³
    Answer

    C. 120 cm³

    Volume = ½ × 6 × 4 × 10 = 120.

  32. A rectangle has perimeter 30 cm and length 9 cm. Its area is

    1. 54 cm²
    2. 81 cm²
    3. 135 cm²
    4. 27 cm²
    Answer

    A. 54 cm²

    Breadth = 15 − 9 = 6, so the area is 54.

  33. The perimeter of a semicircle of radius 7 cm (π = 22/7) is

    1. 44 cm
    2. 36 cm
    3. 29 cm
    4. 22 cm
    Answer

    B. 36 cm

    πr + 2r = 22 + 14 = 36.

  34. Cylinders of equal height have radii in the ratio 2 : 3. Their volumes are in the ratio

    1. 2 : 3
    2. 3 : 2
    3. 8 : 27
    4. 4 : 9
    Answer

    D. 4 : 9

    Volume ∝ r², so the ratio is 4 : 9.

  35. The volume of the frustum of a cone with radii R and r and height h is

    1. (πh/3)(R + r)²
    2. πh(R + r)
    3. (πh/3)(R² + Rr + r²)
    4. (πh/3)(R² − r²)
    Answer

    C. (πh/3)(R² + Rr + r²)

    This is the standard frustum formula.

  36. 1 cubic metre is equal to

    1. 1000 litres
    2. 10,000 litres
    3. 10 litres
    4. 100 litres
    Answer

    A. 1000 litres

    1 m³ = 10⁶ cm³ = 1000 litres.

  37. Which of the statements is/are correct? 1. The volume of a cone is one-third of that of a cylinder with the same base and height. 2. The curved surface area of a cylinder is πr²h.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    The curved surface area is 2πrh.

  38. Which of the statements is/are correct? 1. The volume of a sphere is (4/3)πr³. 2. The surface area of a sphere is 4πr².

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are the standard sphere formulas.

  39. Which of the statements is/are correct? 1. The area of a circle is 2πr. 2. The area of a trapezium is ½ × (sum of parallel sides) × height.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    2πr is the circumference; the area is πr².

  40. Which of the statements is/are correct? 1. The total surface area of a cube is 4a². 2. The volume of a cube is 6a³.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    TSA = 6a² and volume = a³.

  41. Which of the statements is/are correct? 1. The slant height of a cone is l = √(r² + h²). 2. The curved surface area of a cone is πrh.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    The curved surface area is πrl with the slant height l.

  42. Which of the statements is/are correct? 1. If every linear dimension of a solid is doubled, the surface area becomes 4 times. 2. In the same case, the volume becomes 8 times.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Area scales as k² and volume as k³.

  43. Which of the statements is/are correct? 1. The curved surface area of a hemisphere is 2πr². 2. The total surface area of a hemisphere is 2πr².

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    The total surface area includes the flat top and is 3πr².

  44. Match the solid with its volume: (a) cylinder, (b) cone, (c) sphere. The correct formulas in order are

    1. πr²h, (4/3)πr³, ⅓πr²h
    2. πr²h, ⅓πr²h, (4/3)πr³
    3. ⅓πr²h, πr²h, 4πr²
    4. 2πrh, πrl, 4πr²
    Answer

    B. πr²h, ⅓πr²h, (4/3)πr³

    These are the standard volume formulas.

  45. A parallelogram has base 12 cm and height 7 cm. Its area is

    1. 19 cm²
    2. 42 cm²
    3. 168 cm²
    4. 84 cm²
    Answer

    D. 84 cm²

    Area = base × height = 84.

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