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About Mensuration — Class 8 Olympiad

Calculate surface area and volume of cubes, cuboids, cylinders; solve composite solid problems. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 11). On this page you can practice 50 questions across three difficulty levels — 20 easy, 10 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Mensuration — solved examples for Class 8 Olympiad

Example 1easy

A water tank in the shape of a cuboid has dimensions 5 m × 4 m × 3 m. Water is poured into it at a rate of 200 liters per minute. How long will it take to fill the tank completely? (1 m³ = 1000 liters)
  1. A)2 hours
  2. B)3 hours
  3. C)4 hours
  4. D)5 hours

Step-by-step solution

  1. Calculate the volume of the cuboidal tank: Volume = length × width × height = 5 m × 4 m × 3 m = 60 m³.
  2. Convert the volume from cubic meters to liters: Since 1 m³ = 1000 liters, 60 m³ = 60 × 1000 = 60,000 liters.
  3. Calculate the time required to fill the tank: Time = Total volume / Flow rate = 60,000 liters / 200 liters/minute = 300 minutes.
  4. Convert minutes to hours: Time = 300 minutes / 60 minutes/hour = 5 hours.

Answer: 5 hours

Example 2medium

A solid metallic cuboid of dimensions 10 cm × 8 cm × 6 cm has a cylindrical hole of radius 2 cm drilled through its center, parallel to its 10 cm length, passing from one 8 cm × 6 cm face to the opposite 8 cm × 6 cm face. What is the total surface area of the resulting solid?
  1. A) (376 + 20π) cm²
  2. B) (376 + 32π) cm²
  3. C) (376 + 40π) cm²
  4. D) (376 - 8π) cm²

Step-by-step solution

  1. Calculate the original surface area of the cuboid: SA_orig = 2(lb + bh + hl) = 2(10×8 + 8×6 + 6×10) = 2(80 + 48 + 60) = 2(188) = 376 cm².
  2. The cylindrical hole removes two circular areas from the faces it passes through. Radius r = 2 cm. Area removed = 2 × (πr²) = 2 × π(2²) = 8π cm².
  3. The hole adds the curved surface area of the cylinder to the solid. The height of the cylinder is the length of the cuboid along which the hole is drilled, which is 10 cm. Curved surface area added = 2πrh = 2π(2)(10) = 40π cm².
  4. The total surface area of the new solid is SA_orig - Area removed + Area added = 376 - 8π + 40π = (376 + 32π) cm².

Answer: (376 + 32π) cm²

Example 3hard

A solid metallic cube has a peculiar property: its total surface area (in cm²) is numerically equal to half its volume (in cm³). If this cube is then melted and recast into 8 identical smaller cubes, what is the total surface area of one of these smaller cubes?
  1. A)108 cm²
  2. B)144 cm²
  3. C)192 cm²
  4. D)216 cm²

Step-by-step solution

  1. Let the side length of the large cube be 'a'. Its total surface area is 6a² and its volume is a³.
  2. According to the problem, 6a² = (1/2)a³. Since 'a' cannot be zero, we can divide by a²: 6 = a/2, which means a = 12 cm.
  3. The volume of the large cube is a³ = 12³ = 1728 cm³. When recast into 8 identical smaller cubes, the volume of each smaller cube is 1728 / 8 = 216 cm³.
  4. Let the side length of a smaller cube be 's'. Then s³ = 216, so s = ³√216 = 6 cm. The total surface area of one smaller cube is 6s² = 6 × 6² = 6 × 36 = 216 cm².

Answer: 216 cm²

Practice questions on Mensuration

  1. Q1.easy

    A solid wooden cube with side length 4 cm is painted red on all its faces. It is then cut into smaller cubes of side length 1 cm. How many of these smaller cubes have exactly two faces painted red?
    1. A)8
    2. B)12
    3. C)24
    4. D)36
    Show answer

    Answer: 24

    Hint: Cubes with exactly two painted faces are located along the edges of the original large cube, but not at the corners.

  2. Q2.easy

    A cylindrical container has a radius 'r' and height 'h'. If its radius is doubled and its height is halved, what is the ratio of the new volume to the original volume?
    1. A)1 : 1
    2. B)2 : 1
    3. C)4 : 1
    4. D)1 : 2
    Show answer

    Answer: 2 : 1

    Hint: Write down the formula for the volume of a cylinder. Then substitute the new dimensions and find the ratio.

  3. Q3.easy

    A solid metallic cuboid measures 10 cm × 8 cm × 6 cm. A cylindrical hole of radius 2 cm is drilled completely through its 10 cm dimension. What is the volume of the remaining solid? (Use π = 22/7)
    1. A)354.29 cm³
    2. B)379.43 cm³
    3. C)404.57 cm³
    4. D)429.71 cm³
    Show answer

    Answer: 354.29 cm³

    Hint: The length of the cylindrical hole will be the same as the dimension through which it is drilled. Calculate the cuboid's volume and subtract the cylinder's volume.

  4. Q4.medium

    Water flows through a cylindrical pipe of internal diameter 7 cm at a speed of 25 cm/second. This water fills a cuboidal tank with a base of 2 m × 3 m. How long will it take for the water level in the tank to rise by 38.5 cm? (Use π = 22/7)
    1. A) 25 minutes
    2. B) 35 minutes
    3. C) 40 minutes
    4. D) 50 minutes
    Show answer

    Answer: 40 minutes

    Hint: Calculate the total volume of water required in the tank. Then determine the rate of water flow from the cylindrical pipe. Ensure all units are consistent.

  5. Q5.medium

    A solid metallic cube of edge 9 cm is melted and recast into 'n' identical smaller cubes, each with an edge length of 3 cm. If 'S_orig' is the surface area of the original cube and 'S_total_new' is the total surface area of all 'n' smaller cubes combined, what is the ratio S_total_new / S_orig?
    1. A) 2 : 1
    2. B) 3 : 1
    3. C) 4 : 1
    4. D) 5 : 1
    Show answer

    Answer: 3 : 1

    Hint: The volume remains constant during melting and recasting. Determine the number of smaller cubes first, then compare surface areas.

  6. Q6.medium

    A wooden box with external dimensions 15 cm × 10 cm × 8 cm has a uniform thickness of 1 cm. If the cost of wood is ₹5 per cm³, what is the total cost of the wood used to make the box?
    1. A) ₹2500
    2. B) ₹2750
    3. C) ₹2880
    4. D) ₹3000
    Show answer

    Answer: ₹2880

    Hint: The volume of the wood is the difference between the external and internal volumes. Remember to account for thickness on both sides of each dimension.

  7. Q7.hard

    A rectangular prism (cuboid) has integer side lengths. The sum of the lengths of its three distinct edges is 12 cm. The sum of the areas of its three distinct faces is 47 cm². What is the volume of the cuboid?
    1. A)48 cm³
    2. B)54 cm³
    3. C)60 cm³
    4. D)64 cm³
    Show answer

    Answer: 60 cm³

    Hint: Let the side lengths be l, b, h. Formulate equations for the given sums and try to find integer values for l, b, h that satisfy them.

  8. Q8.hard

    A solid right circular cylinder has its height equal to its diameter. If its total surface area is 36π cm², what is its volume?
    1. A)6π√6 cm³
    2. B)12π√6 cm³
    3. C)18π√6 cm³
    4. D)24π√6 cm³
    Show answer

    Answer: 12π√6 cm³

    Hint: Express all cylinder formulas in terms of a single variable (e.g., radius 'r') using the given relationship between height and diameter.

  9. Q9.hard

    A solid metallic cube of side 10 cm has a cylindrical hole of radius 3 cm drilled completely through its center, from one face to the opposite face. What is the volume of the remaining solid? (Use π = 3.14)
    1. A)717.4 cm³
    2. B)720.8 cm³
    3. C)734.6 cm³
    4. D)742.2 cm³
    Show answer

    Answer: 717.4 cm³

    Hint: Calculate the volume of the cube and the volume of the cylindrical hole. The volume of the remaining solid is their difference.

These are 9 of the 50 questions available for Mensuration. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.