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About Mensuration (3D Solids) — Class 10 ICSE

Calculate surface area and volume of cylinders, cones, spheres, and combined solids. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 6). On this page you can practice 56 questions across three difficulty levels — 20 easy, 20 medium, and 16 hard — each with a visual step-by-step solution, plus a timed 33-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Mensuration (3D Solids)

  • Introduction to 3D Solids: Surface Area & Volume
  • Cylinders: Surface Area and Volume
  • Cones and Spheres: Surface Area and Volume
  • Combined Solids: Surface Area and Volume
  • Conservation of Volume & Exam Strategies

Interactive lesson · about 15 minutes · checkpoint question after every unit

Mensuration (3D Solids) — solved examples for Class 10 ICSE

Example 1easy

Which of the following statements best describes the 'volume' of a 3D solid?
  1. A)The total area of all its surfaces.
  2. B)The length of its longest side.
  3. C)The amount of space it occupies or its capacity.
  4. D)The perimeter of its base.

Step-by-step solution

  1. Volume is a measure of the three-dimensional space occupied by a solid object or a fluid.
  2. It quantifies the capacity of the solid, unlike surface area which measures the outer boundary.

Answer: The amount of space it occupies or its capacity.

Example 2medium

A cylindrical water tank has a diameter of 7 m and a height of 10 m. If 1 m³ of water weighs 1000 kg, what is the total weight of water (in tonnes) the tank can hold when full? (Use π = 22/7)
  1. A)38.5 tonnes
  2. B)385 tonnes
  3. C)3850 tonnes
  4. D)38500 tonnes

Step-by-step solution

  1. Radius of the tank, r = Diameter / 2 = 7 m / 2 = 3.5 m.
  2. Volume of the cylindrical tank, V = πr²h = (22/7) × (3.5)² × 10 m³.
  3. V = (22/7) × 12.25 × 10 m³ = 22 × 1.75 × 10 m³ = 385 m³.
  4. Weight of water = Volume × Weight per m³ = 385 m³ × 1000 kg/m³ = 385000 kg. Convert kg to tonnes (1 tonne = 1000 kg): 385000 kg / 1000 = 385 tonnes.

Answer: 385 tonnes

Example 3hard

A hollow cylindrical pipe is made of metal. Its outer diameter is 14 cm and its inner diameter is 10 cm. The height of the pipe is 21 cm. If 1 cm³ of metal weighs 8 grams, what is the total weight of the pipe? (Take π = 22/7)
  1. A)2244 grams
  2. B)2376 grams
  3. C)2112 grams
  4. D)2068 grams

Step-by-step solution

  1. Outer radius (R) = 14/2 = 7 cm. Inner radius (r) = 10/2 = 5 cm. Height (h) = 21 cm.
  2. Volume of metal = Volume of outer cylinder - Volume of inner cylinder = πR²h - πr²h = πh(R² - r²)
  3. Volume of metal = (22/7) × 21 × (7² - 5²) = 22 × 3 × (49 - 25) = 66 × 24 = 1584 cm³.
  4. Total weight of the pipe = Volume of metal × Weight per cm³ = 1584 cm³ × 8 g/cm³ = 12672 grams.

Answer: 2112 grams

Practice questions on Mensuration (3D Solids)

  1. Q1.easy

    A student needs to calculate the area of the curved surface of a cylindrical water tank to paint it from the outside. Which formula should they use?
    1. A)πr²h
    2. B)2πr(r + h)
    3. C)2πrh
    4. D)4πr²
    Show answer

    Answer: 2πrh

    Hint: Consider which formula represents only the lateral surface of a cylinder, excluding its top and bottom circular bases.

  2. Q2.easy

    Ravi is calculating the Total Surface Area (TSA) of a cone with radius 'r', height 'h', and slant height 'l'. He uses the formula TSA = πrl. What mistake did Ravi make?
    1. A)He used the formula for the volume of a cone.
    2. B)He forgot to add the area of the circular base.
    3. C)He used the formula for the curved surface area of a cylinder.
    4. D)The correct formula should be TSA = πr²l.
    Show answer

    Answer: He forgot to add the area of the circular base.

    Hint: Remember that Total Surface Area includes ALL surfaces that make up the solid. A cone has a curved surface and a base.

  3. Q3.easy

    If the radius of a sphere is doubled, how many times will its volume increase?
    1. A)2 times
    2. B)4 times
    3. C)8 times
    4. D)16 times
    Show answer

    Answer: 8 times

    Hint: Recall the formula for the volume of a sphere and how the radius term is powered.

  4. Q4.medium

    A conical heap of rice has a diameter of 24 m and a volume of 1232 m³. What is the approximate height of the heap? (Use π = 22/7)
    1. A)7 m
    2. B)8 m
    3. C)9 m
    4. D)10 m
    Show answer

    Answer: 8 m

    Hint: Use the formula for the volume of a cone and rearrange it to find the height, given the volume and radius.

  5. Q5.medium

    A solid sphere and a solid hemisphere have the same radius 'r'. What is the ratio of their total surface areas?
    1. A)2:1
    2. B)3:2
    3. C)4:3
    4. D)4:1
    Show answer

    Answer: 4:3

    Hint: Recall the formulas for the total surface area of a sphere and a solid hemisphere. Remember a solid hemisphere includes its base area.

  6. Q6.medium

    A solid toy is in the form of a right circular cylinder with a conical top. The radius of both the cylinder and the cone is 7 cm. If the height of the cylinder is 1 cm and the height of the cone is 3 cm, what is the total volume of the toy? (Use π = 22/7)
    1. A)308 cm³
    2. B)330 cm³
    3. C)362 cm³
    4. D)396 cm³
    Show answer

    Answer: 308 cm³

    Hint: Calculate the volume of the cylindrical part and the conical part separately, then add them to find the total volume of the toy.

  7. Q7.hard

    A solid metallic sphere of radius 12 cm is melted and recast into a number of smaller identical cones, each of radius 3 cm and height 4 cm. What is the number of cones formed? (Assume no loss of material during melting)
    1. A)144
    2. B)96
    3. C)192
    4. D)72
    Show answer

    Answer: 192

    Hint: The total volume of the metallic sphere must be equal to the sum of the volumes of all the smaller cones. Equate the volumes to find the number of cones.

  8. Q8.hard

    A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 6 cm. Find the total surface area of the toy. (Take π = 3.14)
    1. A)94.2 cm²
    2. B)103.62 cm²
    3. C)84.78 cm²
    4. D)113.04 cm²
    Show answer

    Answer: 94.2 cm²

    Hint: The total surface area of the toy will be the sum of the curved surface area of the cone and the curved surface area of the hemisphere. Remember to find the slant height of the cone first.

  9. Q9.hard

    If the radius of a sphere is doubled, which of the following statements is TRUE?
    1. A)Its surface area becomes 2 times and its volume becomes 4 times.
    2. B)Its surface area becomes 4 times and its volume becomes 6 times.
    3. C)Its surface area becomes 4 times and its volume becomes 8 times.
    4. D)Its surface area becomes 8 times and its volume becomes 4 times.
    Show answer

    Answer: Its surface area becomes 4 times and its volume becomes 8 times.

    Hint: Recall the formulas for the surface area and volume of a sphere in terms of its radius. Consider how squaring and cubing the radius affects these values when the radius is doubled.

These are 9 of the 56 questions available for Mensuration (3D Solids). Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.