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About Surface Areas & Volumes — Class 9 CBSE

Calculate surface area and volume of cubes, cuboids, cylinders, cones, and spheres. This topic is part of the CBSE Class 9 mathematics syllabus (chapter: Chapter 13). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Surface Areas & Volumes

  • Introduction to Surface Areas & Volumes: Cubes and Cuboids
  • Exploring Cylinders: Curved and Total Surface Areas, and Volume
  • Cones: Slant Height, Surface Area, and Volume
  • Spheres and Hemispheres: Curved and Total Surface Areas, and Volume
  • Summary, Combined Solids, and Practice Strategies

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

Surface Areas & Volumes — solved examples for Class 9 CBSE

Example 1easy

A room's walls and ceiling need to be painted. The floor is not to be painted. The room is in the shape of a cuboid with length 'l', breadth 'b', and height 'h'. Which formula represents the total area to be painted?
  1. A)2(lb + bh + hl)
  2. B)2(bh + hl)
  3. C)2h(l + b) + lb
  4. D)lb + bh + hl

Step-by-step solution

  1. The area of the four walls is the Lateral Surface Area (LSA) of the cuboid, which is 2h(l + b).
  2. The ceiling is one of the faces with dimensions length (l) and breadth (b). Its area is lb.
  3. Since the floor is not painted, the total area to be painted is the sum of the LSA and the area of the ceiling.
    TotalArea=LSA+Areaofceiling=2h(l+b)+lbTotal Area = LSA + Area of ceiling = 2h(l + b) + lb

Answer: 2h(l + b) + lb

Example 2medium

A cuboidal water tank is 6 m long, 5 m wide, and 4.5 m deep. If only 2/3 of the tank is filled with water, what is the volume of water in the tank in litres?
  1. A)A) 135000 litres
  2. B)B) 100000 litres
  3. C)C) 90000 litres
  4. D)D) 162000 litres

Step-by-step solution

  1. Calculate the total volume of the cuboidal tank:
    Volume=length×breadth×height=6m×5m×4.5m=135m3Volume = length × breadth × height = 6 m × 5 m × 4.5 m = 135 m³
  2. The tank is filled to 2/3 of its capacity. Calculate the volume of water:
    Volumeofwater=(2/3)×135m3=90m3Volume of water = (2/3) × 135 m³ = 90 m³
  3. Convert the volume of water from cubic metres to litres (1 m³ = 1000 litres):
    90m3=90×1000litres=90000litres90 m³ = 90 × 1000 litres = 90000 litres

Answer: C) 90000 litres

Example 3hard

A godown is 12 m long, 9 m wide, and 6 m high. It is to store cuboidal boxes, each measuring 1.5 m × 1 m × 0.8 m. If the cost of painting the inner walls and ceiling of the godown is ₹15 per square meter, find the maximum number of boxes that can be stored and the total cost of painting.
  1. A)540 boxes, ₹5400
  2. B)504 boxes, ₹5400
  3. C)540 boxes, ₹4860
  4. D)504 boxes, ₹4860

Step-by-step solution

  1. Calculate the maximum number of boxes that can be stored by dividing the dimensions of the godown by the dimensions of a box, considering all possible orientations. For a godown (L=12, B=9, H=6) and box (l=1.5, b=1, h=0.8):
  2. Arrangement 1: (12/1.5) × (9/1) × (6/0.8) = 8 × 9 × 7 = 504 boxes. (Note: 6/0.8 = 7.5, so 7 boxes fit)
  3. Arrangement 2: (12/0.8) × (9/1.5) × (6/1) = 15 × 6 × 6 = 540 boxes. (Other arrangements yield equal or fewer boxes, so 540 is max).
  4. Area for painting (inner walls and ceiling) = Lateral Surface Area + Area of ceiling = 2(L+B)H + LB = 2(12+9)6 + (12×9) = 2(21)6 + 108 = 252 + 108 = 360 m². Cost of painting = 360 m² × ₹15/m² = ₹5400.

Answer: 540 boxes, ₹5400

Practice questions on Surface Areas & Volumes

  1. Q1.easy

    If the side length of a cube is doubled, how many times does its lateral surface area increase?
    1. A)2 times
    2. B)3 times
    3. C)4 times
    4. D)8 times
    Show answer

    Answer: 4 times

    Hint: Recall the formula for the lateral surface area of a cube and see how it changes when the side length is multiplied by 2.

  2. Q2.easy

    A company wants to design a label for a cylindrical can of juice. The label will cover only the curved part of the can, from top to bottom. Which of the following geometric areas should the company calculate for the label?
    1. A)Total Surface Area
    2. B)Volume
    3. C)Curved Surface Area
    4. D)Area of the base
    Show answer

    Answer: Curved Surface Area

    Hint: Consider which part of the cylinder is being covered by the label. Does it include the top or bottom circular faces?

  3. Q3.easy

    Ravi calculated the total surface area of a cuboid with length 5 cm, breadth 3 cm, and height 2 cm, and got an answer of 62 cm². His teacher said his calculation was correct but mentioned he might have made a conceptual mistake in how he arrived at the formula. Which of the following statements likely describes Ravi's conceptual mistake?
    1. A)He confused surface area with volume.
    2. B)He used the formula for the Lateral Surface Area instead of Total Surface Area.
    3. C)He incorrectly assumed all faces were squares.
    4. D)He calculated the sum of areas of only three unique faces and multiplied by 2, which is the correct approach.
    Show answer

    Answer: He calculated the sum of areas of only three unique faces and multiplied by 2, which is the correct approach.

    Hint: Think about the formula for the total surface area of a cuboid and how it's derived from the areas of its faces. The question states his calculation was correct but implies a conceptual misunderstanding of the formula's derivation.

  4. Q4.medium

    The sides of two cubes are in the ratio 2:3. What is the ratio of their total surface areas?
    1. A)A) 2:3
    2. B)B) 4:9
    3. C)C) 8:27
    4. D)D) 1:3
    Show answer

    Answer: B) 4:9

    Hint: Remember that the total surface area of a cube is proportional to the square of its side length.

  5. Q5.medium

    A cylindrical pillar has a diameter of 50 cm and a height of 3.5 m. Find the cost of painting the curved surface of the pillar at the rate of ₹12.50 per m². (Use π = 22/7)
    1. A)A) ₹68.75
    2. B)B) ₹687.50
    3. C)C) ₹137.50
    4. D)D) ₹1375.00
    Show answer

    Answer: A) ₹68.75

    Hint: First, ensure all dimensions are in the same unit (metres). Then calculate the curved surface area of the cylinder and multiply by the painting rate.

  6. Q6.medium

    If the volume of a sphere is 4851 cm³, find its radius. (Use π = 22/7)
    1. A)A) 7 cm
    2. B)B) 10.5 cm
    3. C)C) 14 cm
    4. D)D) 21 cm
    Show answer

    Answer: B) 10.5 cm

    Hint: Use the formula for the volume of a sphere and solve for the radius (r). Remember V = (4/3)πr³.

  7. Q7.hard

    Three identical cubes, each of side 'a', are joined end-to-end to form a single cuboid. Find the ratio of the total surface area of the resulting cuboid to the sum of the total surface areas of the three original cubes.
    1. A)3/5
    2. B)5/7
    3. C)6/11
    4. D)7/9
    Show answer

    Answer: 7/9

    Hint: When cubes are joined, their internal faces merge, reducing the total exposed surface area. Calculate TSA for individual cubes and the resulting cuboid carefully.

  8. Q8.hard

    A well with 10 m inside diameter is dug 14 m deep. Earth taken out of it is spread all around to a width of 5 m to form an embankment. Find the height of the embankment. (Use π = 22/7)
    1. A)3.5 m
    2. B)14/3 m
    3. C)7 m
    4. D)10/3 m
    Show answer

    Answer: 14/3 m

    Hint: The volume of earth removed from the well is equal to the volume of the embankment. The embankment is a hollow cylinder, so consider its inner and outer radii.

  9. Q9.hard

    The sum of the length, breadth, and height of a cuboid is 19 cm. Its diagonal is 11 cm. Find its total surface area.
    1. A)121 cm²
    2. B)240 cm²
    3. C)361 cm²
    4. D)480 cm²
    Show answer

    Answer: 240 cm²

    Hint: Recall the formula for the diagonal of a cuboid (d = √(l² + b² + h²)) and the algebraic identity for (l+b+h)².

These are 9 of the 60 questions available for Surface Areas & Volumes. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.