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About 3D Geometry — Class 9 IB

Calculate surface area and volume of pyramids, cones, and spheres; solve composite solid problems. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 7). 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.

3D Geometry — solved examples for Class 9 IB

Example 1easy

Which of the following statements about the volume of a sphere is correct?
  1. A)If the radius of a sphere is doubled, its volume becomes eight times the original volume.
  2. B)The volume of a sphere is directly proportional to its radius.
  3. C)Doubling the radius of a sphere halves its volume.
  4. D)The volume of a sphere depends on its surface area, not directly on its radius.

Step-by-step solution

  1. The formula for the volume of a sphere is V = (4/3)πr³.
  2. If the radius 'r' is doubled to '2r', the new volume V' = (4/3)π(2r)³ = (4/3)π(8r³) = 8V. Thus, statement A is correct.
  3. The volume is proportional to r³, not r, so option B is incorrect. Doubling the radius increases the volume by a factor of 8, not halves it, so option C is incorrect. The volume directly depends on the radius, so D is incorrect.

Answer: If the radius of a sphere is doubled, its volume becomes eight times the original volume.

Example 2medium

A spherical water tank has a diameter of 2.1 meters. What is its capacity in liters? (Use π = 22/7 and 1 m³ = 1000 liters)
  1. A)2425.5 liters
  2. B)4851 liters
  3. C)9702 liters
  4. D)1212.75 liters

Step-by-step solution

  1. Diameter = 2.1 m, so radius (r) = 2.1 / 2 = 1.05 m.
  2. Volume of sphere (V) = (4/3)πr³ = (4/3) × (22/7) × (1.05)³ m³
  3. V = (4/3) × (22/7) × 1.157625 = 4.851 m³
  4. Convert to liters: 4.851 m³ × 1000 liters/m³ = 4851 liters.

Answer: 4851 liters

Example 3hard

A solid right square pyramid has a base side length of 10 cm. If all its lateral faces are equilateral triangles, what is the volume of the pyramid?
  1. A)(250√3)/3 cm³
  2. B)(500√2)/3 cm³
  3. C)(100√5)/3 cm³
  4. D)500 cm³

Step-by-step solution

  1. The base side length (a) = 10 cm. Since lateral faces are equilateral triangles, the slant height (l) of the pyramid is equal to the side length of the equilateral triangle, which is 10 cm.
  2. The apothem (distance from center of base to midpoint of a side) of the square base is a/2 = 10/2 = 5 cm.
  3. Using the Pythagorean theorem for the vertical height (h), apothem, and slant height: h² + (apothem)² = l². So, h² + 5² = 10². h² + 25 = 100. h² = 75. h = √75 = 5√3 cm. (Correction: This is incorrect. If lateral faces are equilateral triangles, their height is the slant height of the pyramid. The slant height of the pyramid is the altitude of the equilateral triangle face, which is (√3/2)*side. The *edge* of the pyramid is 10 cm. The slant height 'l' is the height of the triangular face, which is (√3/2)*10 = 5√3 cm. Apothem 'x' = 10/2 = 5 cm. Then h² + x² = l². h² + 5² = (5√3)². h² + 25 = 75. h² = 50. h = 5√2 cm.)
  4. Volume of pyramid = (1/3) × Base Area × Height = (1/3) × (10 cm)² × (5√2 cm) = (1/3) × 100 × 5√2 = (500√2)/3 cm³.

Answer: (500√2)/3 cm³

Practice questions on 3D Geometry

  1. Q1.easy

    A right circular cone has a base radius of 7 cm and a height of 24 cm. What is its volume? (Use π = 22/7)
    1. A)1232 cm³
    2. B)154 cm³
    3. C)3696 cm³
    4. D)410.67 cm³
    Show answer

    Answer: 1232 cm³

    Hint: Remember the formula for the volume of a cone, and carefully substitute the given values.

  2. Q2.easy

    Ravi calculated the total surface area of a square-based pyramid with a base side length of 6 cm and a slant height of 5 cm. His steps are:
    1. Area of base = 6 cm × 6 cm = 36 cm²
    2. Area of one triangular face = (1/2) × 6 cm × 5 cm = 15 cm²
    3. Total surface area = 36 cm² + 15 cm² = 51 cm²
    What mistake did Ravi make?
    1. A)He used the incorrect formula for the area of the base.
    2. B)He used the incorrect formula for the area of a triangular face.
    3. C)He only added the area of one triangular face instead of all four.
    4. D)He confused slant height with vertical height.
    Show answer

    Answer: He only added the area of one triangular face instead of all four.

    Hint: A square-based pyramid has four triangular faces. Consider how many of these faces contribute to the total surface area.

  3. Q3.easy

    When calculating the lateral surface area of a cone or a pyramid, we use the slant height (l) instead of the perpendicular height (h). Why is this necessary?
    1. A)The slant height is always shorter than the perpendicular height, simplifying calculations.
    2. B)The lateral faces of cones and pyramids are flat, and their actual height along the surface is the slant height.
    3. C)The perpendicular height is only used for volume calculations, not surface area.
    4. D)Using slant height ensures the formula includes the radius, which is crucial for surface area.
    Show answer

    Answer: The lateral faces of cones and pyramids are flat, and their actual height along the surface is the slant height.

    Hint: Consider what 'lateral surface area' represents – the area of the sloping sides. The slant height measures the height along these sloping surfaces.

  4. Q4.medium

    A conical tent has a volume of 1232 m³ and its base area is 154 m². What is the height of the tent?
    1. A)12 m
    2. B)24 m
    3. C)32 m
    4. D)48 m
    Show answer

    Answer: 24 m

    Hint: Recall the formula for the volume of a cone, which relates volume, base area, and height. You can rearrange it to find the height.

  5. Q5.medium

    A craftsman needs to paint the external surface of a solid square-based pyramid, including its base. The base has a side length of 10 cm and the perpendicular height of the pyramid is 12 cm. If the paint costs ₹0.50 per cm², what is the total cost of painting? (Assume exact calculations, no wastage)
    1. A)₹130
    2. B)₹260
    3. C)₹340
    4. D)₹420
    Show answer

    Answer: ₹340

    Hint: You need to find the slant height first using Pythagoras theorem. Then calculate the total surface area (base area + lateral surface area) and multiply by the cost per cm².

  6. Q6.medium

    If the radius of a sphere is doubled, what will be the effect on its volume?
    1. A)The volume will be doubled.
    2. B)The volume will be quadrupled.
    3. C)The volume will be 8 times its original volume.
    4. D)The volume will be 16 times its original volume.
    Show answer

    Answer: The volume will be 8 times its original volume.

    Hint: Recall the formula for the volume of a sphere and consider how 'r³' changes when 'r' becomes '2r'.

  7. Q7.hard

    A conical tent has a volume of 1232 m³. The area of its base is 154 m². Find the slant height of the tent.
    1. A)15 m
    2. B)20 m
    3. C)25 m
    4. D)28 m
    Show answer

    Answer: 25 m

    Hint: First, use the base area to find the radius of the cone. Then, use the volume and radius to find the vertical height. Finally, apply the Pythagorean theorem to find the slant height.

  8. Q8.hard

    A solid metallic sphere of radius R is melted and recast into 'n' identical smaller solid spheres of radius r. If the total surface area of the 'n' smaller spheres is twice the surface area of the original sphere, what is the ratio R/r?
    1. A)2
    2. B)4
    3. C)√2
    4. D)2√2
    Show answer

    Answer: 2

    Hint: Conservation of volume applies when melting and recasting. Establish equations for volume and surface area, then substitute to find the relationship between R and r.

  9. Q9.hard

    A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. (Use π = 3.14)
    1. A)8π cm³
    2. B)10.46 cm³
    3. C)16.75 cm³
    4. D)25.12 cm³
    Show answer

    Answer: 25.12 cm³

    Hint: Calculate the volume of the hemispherical part and the conical part separately. Remember that the radius of the cone is the same as the radius of the hemisphere.

These are 9 of the 60 questions available for 3D Geometry. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.