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Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 12

Mensuration: Perimeter, Area, Surface Area and Volume

What to remember

  • Perimeter is the length around a closed figure (unit: cm, m). Area is the surface it covers (unit: cm², m²). Volume is the space a solid fills (unit: cm³, m³).
  • Cube: TSA = 6a², V = a³. Cuboid: TSA = 2(lb + bh + hl), V = lbh. Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h.
  • 1 m³ = 1000 litres and 1 cm³ = 1 mL. Use π = 22/7 unless told otherwise.

Perimeter and area of plane figures

FigurePerimeterArea
Square (side a)4aa²
Rectangle (l, b)2(l + b)l × b
Triangle (base b, height h)sum of three sides½ × b × h
Parallelogram (base b, height h)2(sum of adjacent sides)b × h
Rhombus (diagonals d₁, d₂)4 × side½ × d₁ × d₂
Trapezium (parallel sides a, b; height h)sum of sides½ × (a + b) × h
Circle (radius r)Circumference = 2πrπr²
  • Equilateral triangle of side a: perimeter 3a, area (√3 ÷ 4) × a².
  • Right-angled triangle: area = ½ × product of the two sides forming the right angle.
  • Heron's formula: area = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.
  • Diagonal of a square = a√2. Diagonal of a rectangle = √(l² + b²).
  • Diameter d = 2r. Circumference = πd.
  • Area of a path (ring): area of the outer figure − area of the inner figure.
  • Area of a semicircle = ½πr². Its perimeter = πr + 2r.

Examples:

  • A rectangle 12 m by 8 m has perimeter 2 × 20 = 40 m and area 96 m².
  • A square of perimeter 36 cm has side 9 cm and area 81 cm².
  • A circle of radius 7 cm has circumference 2 × 22/7 × 7 = 44 cm and area 22/7 × 49 = 154 cm².
  • A triangle with base 10 cm and height 6 cm has area 30 cm².
  • A trapezium with parallel sides 8 cm and 12 cm and height 5 cm has area ½ × 20 × 5 = 50 cm².

Units and conversions

QuantityConversion
1 m100 cm
1 m²10,000 cm²
1 hectare10,000 m²
1 m³1,000,000 cm³
1 litre1000 cm³
1 m³1000 litres
1 cm³1 mL
1 kilolitre1 m³

Area uses square units and volume uses cubic units. Convert all lengths to the same unit before calculating.

Surface area and volume of solids

Lateral (or curved) surface area (LSA/CSA) is the area of the side faces only. Total surface area (TSA) includes the top and bottom too. Volume is the space inside the solid (capacity when it holds a liquid).

SolidLSA / CSATSAVolume
Cube (side a)4a²6a²a³
Cuboid (l, b, h)2h(l + b)2(lb + bh + hl)l × b × h
Cylinder (r, h)2πrh2πr(r + h)πr²h

Other useful results:

  • Diagonal of a cube = a√3. Diagonal of a cuboid = √(l² + b² + h²).
  • Cuboid without a top or open box: TSA = lb + 2h(l + b) (one face missing).
  • Cylinder open at one end: area = 2πrh + πr².
  • Hollow cylinder (pipe): volume of material = π(R² − r²)h, with outer radius R and inner radius r.
  • A cylinder's curved surface opens out to a rectangle of length 2πr and breadth h.
  • Volume of a cuboid = area of base × height. This works for all prisms and cylinders: V = base area × height.
  • Cuboid with square base: l = b.
  • Four walls of a room: 2h(l + b). Area to paint also includes the ceiling but excludes doors and windows.
  • Cost = area (or volume) × rate.

Worked examples:

  • 1. Cube of side 5 cm: TSA = 6 × 25 = 150 cm². Volume = 125 cm³. LSA = 100 cm².
  • 2. Cuboid 10 cm × 6 cm × 4 cm: TSA = 2(60 + 24 + 40) = 248 cm². Volume = 240 cm³. LSA = 2 × 4 × 16 = 128 cm².
  • 3. Cylinder with r = 7 cm and h = 10 cm: CSA = 2 × 22/7 × 7 × 10 = 440 cm². TSA = 2 × 22/7 × 7 × 17 = 748 cm². Volume = 22/7 × 49 × 10 = 1540 cm³.
  • 4. Tank 4 m × 3 m × 2.5 m: Volume = 30 m³ = 30,000 litres.
  • 5. Cube with volume 343 cm³: side = 7 cm (since 7³ = 343), TSA = 6 × 49 = 294 cm².
  • 6. Cuboid with volume 480 cm³, length 10 and breadth 8: height = 480 ÷ 80 = 6 cm.
  • 7. Cylinder with volume 1540 cm³ and r = 7: h = 1540 ÷ (22/7 × 49) = 10 cm.
  • 8. A cube's side is doubled: the surface area becomes 4 times and the volume becomes 8 times.
  • 9. Number of small cubes: a cuboid 12 cm × 6 cm × 4 cm is cut into cubes of side 2 cm. Number = (12 × 6 × 4) ÷ 8 = 36.
  • 10. Cylinder radius doubled, height same: volume becomes 4 times.

Changing shape: When a solid is melted and recast, the volume stays the same. A metal cube of side 6 cm is melted into a cuboid with a base of 9 cm × 4 cm, so height = 216 ÷ 36 = 6 cm.

Filling a tank: Water flowing through a pipe forms a cylinder of length equal to the distance flowed.

Quick comparison

FeatureCubeCuboidCylinder
Faces6 equal squares6 rectangles2 circles and a curved surface
TSA6a²2(lb + bh + hl)2πr(r + h)
Volumea³lbhπr²h
Number of lateral faces441 curved

Classroom angle

Use the "unit cube" method: stack small cubes inside a box to show volume as the count of unit cubes. Cover a box with paper to show surface area. Roll a sheet of paper into a cylinder to show that its curved surface is a rectangle. Always insist on units, and ask children whether the answer is length, area or volume before choosing the formula. Use classroom objects: a duster, a chalk box, a water bottle.

Exam traps

  • Perimeter is in cm, area in cm², volume in cm³; marks are lost for missing or wrong units.
  • LSA of a cube is 4a², not 6a².
  • Radius is half of the diameter. Check which one is given.
  • The area of a triangle has the factor ½; a parallelogram does not.
  • Convert units first: 1 m³ = 1000 litres, not 100.
  • The cylinder's TSA formula includes both circular ends: 2πr(r + h). CSA does not.
  • Doubling the side of a cube increases the surface area 4 times and the volume 8 times, not 2 times.
  • For a trapezium, the height is the perpendicular distance, not the slant side.

One-liners

  • Area of a rectangle = l × b.
  • Area of a triangle = ½ × base × height.
  • Area of a circle = πr².
  • Circumference = 2πr.
  • Cube: TSA = 6a² and volume = a³.
  • Cuboid: TSA = 2(lb + bh + hl) and volume = lbh.
  • Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h.
  • Diagonal of a cube = a√3.
  • 1 m³ = 1000 litres.
  • 1 cm³ = 1 mL.
  • 1 hectare = 10,000 m².
  • When a solid is recast, its volume does not change.

Practice questions

  1. The total surface area of a cube of side a is

    1. a³
    2. 6a²
    3. 4a²
    4. 12a²
    Answer

    B. 6a²

    A cube has 6 square faces of area a².

  2. The volume of a cube of side a is

    1. a³
    2. 3a
    3. a²
    4. 6a²
    Answer

    A. a³

    Volume = side × side × side.

  3. The total surface area of a cuboid is

    1. lbh
    2. 2(lb + bh + hl)
    3. l + b + h
    4. 2h(l + b)
    Answer

    B. 2(lb + bh + hl)

    Three pairs of opposite faces.

  4. The curved surface area of a cylinder is

    1. πr²h
    2. πr²
    3. 2πrh
    4. 2πr(r + h)
    Answer

    C. 2πrh

    The curved surface opens into a rectangle 2πr by h.

  5. The volume of a cylinder is

    1. 2πrh
    2. πrh
    3. 2πr(r + h)
    4. πr²h
    Answer

    D. πr²h

    Volume = base area × height.

  6. The total surface area of a cylinder is

    1. 2πr(r + h)
    2. πr(r + h)
    3. 2πrh
    4. πr²h
    Answer

    A. 2πr(r + h)

    Curved surface plus two circular ends: 2πrh + 2πr².

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

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

    D. 154 cm²

    22/7 × 49 = 154.

  8. The circumference of a circle of radius 7 cm is

    1. 22 cm
    2. 154 cm
    3. 44 cm
    4. 88 cm
    Answer

    C. 44 cm

    2 × 22/7 × 7 = 44.

  9. The area of a triangle with base 10 cm and height 6 cm is

    1. 36 cm²
    2. 30 cm²
    3. 60 cm²
    4. 16 cm²
    Answer

    B. 30 cm²

    ½ × 10 × 6 = 30.

  10. The area of a trapezium with parallel sides 8 cm and 12 cm and height 5 cm is

    1. 100 cm²
    2. 60 cm²
    3. 40 cm²
    4. 50 cm²
    Answer

    D. 50 cm²

    ½ × (8 + 12) × 5 = 50.

  11. The perimeter of a rectangle 12 m by 8 m is

    1. 96 m
    2. 40 m
    3. 48 m
    4. 20 m
    Answer

    B. 40 m

    2(12 + 8) = 40.

  12. A square has perimeter 36 cm. Its area is

    1. 81 cm²
    2. 36 cm²
    3. 144 cm²
    4. 9 cm²
    Answer

    A. 81 cm²

    Side = 9 cm; area = 81.

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

    1. 25 cm²
    2. 125 cm²
    3. 150 cm²
    4. 100 cm²
    Answer

    C. 150 cm²

    6 × 25 = 150.

  14. The volume of a cuboid 10 cm × 6 cm × 4 cm is

    1. 240 cm³
    2. 248 cm³
    3. 120 cm³
    4. 200 cm³
    Answer

    A. 240 cm³

    10 × 6 × 4 = 240.

  15. The total surface area of a cuboid 10 cm × 6 cm × 4 cm is

    1. 124 cm²
    2. 240 cm²
    3. 248 cm²
    4. 200 cm²
    Answer

    C. 248 cm²

    2(60 + 24 + 40) = 248.

  16. The curved surface area of a cylinder with r = 7 cm and h = 10 cm is

    1. 220 cm²
    2. 1540 cm²
    3. 748 cm²
    4. 440 cm²
    Answer

    D. 440 cm²

    2 × 22/7 × 7 × 10 = 440.

  17. The volume of a cylinder with r = 7 cm and h = 10 cm is

    1. 154 cm³
    2. 1540 cm³
    3. 748 cm³
    4. 440 cm³
    Answer

    B. 1540 cm³

    22/7 × 49 × 10 = 1540.

  18. The total surface area of a cylinder with r = 7 cm and h = 10 cm is

    1. 594 cm²
    2. 440 cm²
    3. 748 cm²
    4. 1540 cm²
    Answer

    C. 748 cm²

    2 × 22/7 × 7 × 17 = 748.

  19. A tank is 4 m × 3 m × 2.5 m. How many litres of water can it hold?

    1. 30,000
    2. 300,000
    3. 300
    4. 3,000
    Answer

    A. 30,000

    Volume = 30 m³ and 1 m³ = 1000 litres.

  20. A cube has volume 343 cm³. Its side is

    1. 6 cm
    2. 7 cm
    3. 49 cm
    4. 8 cm
    Answer

    B. 7 cm

    7³ = 343.

  21. A cube has volume 343 cm³. Its total surface area is

    1. 49 cm²
    2. 196 cm²
    3. 343 cm²
    4. 294 cm²
    Answer

    D. 294 cm²

    Side 7; 6 × 49 = 294.

  22. A cuboid has volume 480 cm³, length 10 cm and breadth 8 cm. Its height is

    1. 4 cm
    2. 5 cm
    3. 8 cm
    4. 6 cm
    Answer

    D. 6 cm

    480 ÷ 80 = 6.

  23. The side of a cube is doubled. Its surface area becomes

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

    A. 4 times

    Area depends on a², so doubling gives 4 times.

  24. The side of a cube is doubled. Its volume becomes

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

    B. 8 times

    Volume depends on a³, so doubling gives 8 times.

  25. The radius of a cylinder is doubled and its height stays the same. The volume becomes

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

    C. 4 times

    V = πr²h, so r doubled gives 4 times.

  26. How many cubes of side 2 cm can be cut from a cuboid 12 cm × 6 cm × 4 cm?

    1. 36
    2. 24
    3. 72
    4. 18
    Answer

    A. 36

    288 ÷ 8 = 36.

  27. A metal cube of side 6 cm is melted and recast into a cuboid with base 9 cm × 4 cm. Its height is

    1. 4 cm
    2. 3 cm
    3. 9 cm
    4. 6 cm
    Answer

    D. 6 cm

    216 ÷ 36 = 6.

  28. The diagonal of a cube of side a is

    1. a²
    2. 3a
    3. a√3
    4. a√2
    Answer

    C. a√3

    Diagonal = √(a² + a² + a²) = a√3.

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

    1. 15 cm
    2. 13 cm
    3. 12 cm
    4. 19 cm
    Answer

    B. 13 cm

    √(9 + 16 + 144) = √169 = 13.

  30. 1 cm³ of water equals

    1. 1 litre
    2. 1 mL
    3. 100 mL
    4. 10 mL
    Answer

    B. 1 mL

    1 litre = 1000 cm³, so 1 cm³ = 1 mL.

  31. The area of a rhombus with diagonals 16 cm and 12 cm is

    1. 96 cm²
    2. 48 cm²
    3. 28 cm²
    4. 192 cm²
    Answer

    A. 96 cm²

    ½ × 16 × 12 = 96.

  32. What is the best way to help Class V children understand volume?

    1. Fill a box with unit cubes and count them
    2. Ask them to memorise the formula
    3. Show only the formula on the board
    4. Give only word problems
    Answer

    A. Fill a box with unit cubes and count them

    Counting unit cubes shows volume as the space filled.

  33. What is the first step when a problem gives lengths in both metres and centimetres?

    1. Ignore the smaller unit
    2. Multiply by 100 at the end
    3. Add them as they are
    4. Convert them into the same unit
    Answer

    D. Convert them into the same unit

    All measures must be in the same unit before using a formula.

  34. Statements: 1. The lateral surface area of a cube is 4a². 2. The total surface area of a cube is 6a². Which is/are correct?

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

    D. Both 1 and 2

    Four side faces give 4a²; all six faces give 6a².

  35. Statements: 1. Area is measured in cubic units. 2. Volume is measured in cubic units. Which is/are correct?

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

    D. 2 only

    Area uses square units.

  36. Statements: 1. When a solid is melted and recast, its volume stays the same. 2. When a solid is melted and recast, its surface area stays the same. Which is/are correct?

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

    D. 1 only

    Volume is conserved; surface area usually changes.

  37. Statements: 1. 1 m³ = 1000 litres. 2. 1 m³ = 100 litres. Which is/are correct?

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

    A. 1 only

    1 m³ = 1000 litres.

  38. Statements: 1. The curved surface of a cylinder opens out into a rectangle. 2. The length of that rectangle equals the circumference of the base. Which is/are correct?

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

    D. Both 1 and 2

    The rectangle is 2πr long and h wide.

  39. Statements: 1. Doubling the radius of a circle doubles its area. 2. Doubling the radius of a circle doubles its circumference. Which is/are correct?

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

    D. 2 only

    Area depends on r², so doubling the radius makes the area 4 times.

  40. Match the solid with its volume formula: P. Cube Q. Cuboid R. Cylinder 1. lbh 2. a³ 3. πr²h

    1. P-2, Q-3, R-1
    2. P-3, Q-1, R-2
    3. P-2, Q-1, R-3
    4. P-1, Q-2, R-3
    Answer

    C. P-2, Q-1, R-3

    Cube a³, cuboid lbh, cylinder πr²h.

  41. Match the figure with its area: P. Triangle Q. Parallelogram R. Circle S. Rhombus 1. b × h 2. ½ × b × h 3. πr² 4. ½ × d₁ × d₂

    1. P-2, Q-1, R-4, S-3
    2. P-2, Q-1, R-3, S-4
    3. P-1, Q-2, R-3, S-4
    4. P-2, Q-3, R-1, S-4
    Answer

    B. P-2, Q-1, R-3, S-4

    Standard area formulas.

  42. Match each solid with its total surface area: P. Cube of side 4 cm Q. Cuboid 5 × 4 × 3 cm R. Cylinder r = 7, h = 3 cm 1. 96 cm² 2. 94 cm² 3. 440 cm²

    1. P-1, Q-3, R-2
    2. P-3, Q-2, R-1
    3. P-1, Q-2, R-3
    4. P-2, Q-1, R-3
    Answer

    C. P-1, Q-2, R-3

    6 × 16 = 96; 2(20 + 12 + 15) = 94; 2 × 22/7 × 7 × 10 = 440.

  43. An open cuboidal box of length 10 cm, breadth 8 cm and height 5 cm has outer surface area (top missing) of

    1. 340 cm²
    2. 260 cm²
    3. 80 cm²
    4. 400 cm²
    Answer

    B. 260 cm²

    lb + 2h(l + b) = 80 + 2 × 5 × 18 = 260.

  44. The cost of painting the four walls of a room 5 m × 4 m × 3 m at ₹10 per m² is

    1. ₹900
    2. ₹270
    3. ₹540
    4. ₹600
    Answer

    C. ₹540

    Walls area = 2 × 3 × 9 = 54 m², cost = 540.

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