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Quantitative Aptitude Notes

Mensuration 3D Notes for SSC Exams

Rahul Kumar

SSC Exam Expert & Content Editor

All notes
Updated 15 Oct 2026

In this note

  1. 1.Volume and Surface Area Formulas
  2. 2.Useful Results
  3. 3.Frustum and Other Solids
  4. 4.Ratio and Percentage Changes
  5. 5.Solved Examples
  6. 6.Quick Tips to Solve Faster
  7. 7.Common Mistakes to Avoid
  8. 8.Quick Revision Points
  9. 9.Practice Questions With Answers
  10. 10.How to Practise This Chapter
  11. 11.Related Notes and Practice Papers
  12. 12.Explore More SSC Subjects
  13. 13.Frequently Asked Questions

PDF Notes: Mensuration 3D Notes for SSC Exams

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Mensuration 3D covers the volume and surface area of solid shapes such as the cube, cuboid, cylinder, cone and sphere. One or two questions come from it in each SSC Tier 1 paper and more in Tier 2. These mensuration 3D notes for SSC give all the formulas in one table, with solved examples and practice questions.

Volume and Surface Area Formulas

Solid Volume Surface area
Cube (side a) a cubed TSA = 6 a squared; diagonal = a root 3
Cuboid (l, b, h) l x b x h TSA = 2(lb + bh + hl); diagonal = square root of (l squared + b squared + h squared)
Cylinder (r, h) pi r squared h CSA = 2 pi r h; TSA = 2 pi r (r + h)
Cone (r, h, slant l) 1/3 pi r squared h CSA = pi r l; TSA = pi r (r + l); l squared = r squared + h squared
Sphere (r) 4/3 pi r cubed SA = 4 pi r squared
Hemisphere (r) 2/3 pi r cubed CSA = 2 pi r squared; TSA = 3 pi r squared

Useful Results

  • If a solid is melted and recast, the volume remains the same.
  • If the radius of a sphere becomes n times, the volume becomes n cubed times and the surface area becomes n squared times.
  • Volume of a cone is one-third of the volume of a cylinder with the same radius and height.
  • Volume of 3 cones = volume of 1 cylinder (same base and height).
  • Number of small solids from a large solid = volume of large / volume of small.
  • 1 cubic metre = 1000 litres.

Frustum and Other Solids

A frustum of a cone is the part left when a small cone is cut off from the top of a big cone. Its volume is 1/3 pi h (R squared + Rr + r squared). SSC mostly asks the standard solids, but if a frustum appears, use this formula. For a hollow cylinder, the volume of the material is pi h (R squared - r squared).

Ratio and Percentage Changes

If every edge of a cube is doubled, the surface area becomes 4 times and the volume becomes 8 times. If the radius of a cylinder is doubled while the height remains the same, the volume becomes 4 times.

Solved Examples

Example 1: Find the volume and total surface area of a cube of side 5 cm.
Solution: Volume = 5 cubed = 125 cubic cm. TSA = 6 x 25 = 150 square cm.

Example 2: Find the volume and total surface area of a cuboid of 10 cm x 6 cm x 4 cm.
Solution: Volume = 10 x 6 x 4 = 240 cubic cm. TSA = 2(60 + 24 + 40) = 248 square cm.

Example 3: Find the volume and curved surface area of a cylinder of radius 7 cm and height 10 cm.
Solution: Volume = 22/7 x 49 x 10 = 1,540 cubic cm. CSA = 2 x 22/7 x 7 x 10 = 440 square cm.

Example 4: Find the volume and curved surface area of a cone with radius 7 cm and height 24 cm.
Solution: Slant height = square root of (49 + 576) = 25 cm. Volume = 1/3 x 22/7 x 49 x 24 = 1,232 cubic cm. CSA = 22/7 x 7 x 25 = 550 square cm.

Example 5: Find the volume and surface area of a sphere of radius 21 cm.
Solution: Volume = 4/3 x 22/7 x 21 cubed = 38,808 cubic cm. SA = 4 x 22/7 x 441 = 5,544 square cm.

Example 6: A cuboid of size 12 cm x 9 cm x 6 cm is melted and recast into cubes of side 3 cm. How many cubes are made?
Solution: Volume of cuboid = 648. Volume of one cube = 27. Number of cubes = 648 / 27 = 24.

Quick Tips to Solve Faster

  • Check whether the question asks for curved, lateral or total surface area.
  • Find the slant height of a cone first using the Pythagoras theorem.
  • When a solid is melted, equate the volumes of the old and new solids.
  • Use 22/7 when the radius is a multiple of 7.
  • Remember 1 cubic metre = 1000 litres for tank questions.

Common Mistakes to Avoid

  • Using diameter in place of radius.
  • Mixing the curved surface area and the total surface area.
  • Forgetting the 1/3 in the cone volume.
  • Using the height instead of the slant height in the CSA of a cone.
  • Not squaring or cubing the unit conversion between cm and m.

Quick Revision Points

  • Cylinder: V = pi r squared h; CSA = 2 pi r h.
  • Cone: V = 1/3 pi r squared h; CSA = pi r l.
  • Sphere: V = 4/3 pi r cubed; SA = 4 pi r squared.
  • Hemisphere: CSA = 2 pi r squared and TSA = 3 pi r squared.
  • Recasting: volume remains constant.

Practice Questions With Answers

  1. Find the volume of a cube of side 6 cm.
    Answer: 216 cubic cm (6 cubed).
  2. Find the total surface area of a cube of side 4 cm.
    Answer: 96 square cm (6 x 16).
  3. Find the length of the diagonal of a cube of side 5 cm.
    Answer: 5 root 3 cm (a root 3).
  4. Find the volume of a cylinder of radius 7 cm and height 5 cm.
    Answer: 770 cubic cm (22/7 x 49 x 5).
  5. Find the curved surface area of a cone with radius 6 cm and height 8 cm. Give the answer in terms of pi.
    Answer: 60 pi square cm (l = 10, pi r l).
  6. Find the surface area of a sphere of radius 3 cm in terms of pi.
    Answer: 36 pi square cm (4 pi r squared).
  7. Find the curved surface area of a hemisphere of radius 7 cm.
    Answer: 308 square cm (2 x 22/7 x 49).
  8. The volume of a cuboid is 480 cubic cm. Its length is 10 cm and its breadth is 8 cm. Find the height.
    Answer: 6 cm (480 / 80).
  9. If the radius of a sphere is doubled, how many times does its volume become?
    Answer: 8 times (2 cubed).
  10. How many cubes of side 2 cm can be cut from a cube of side 10 cm?
    Answer: 125 (1000 / 8).

How to Practise This Chapter

  1. Learn the formulas in the table until you can write them without looking.
  2. Solve the solved examples again on your own, without reading the solution.
  3. Attempt the practice questions above with a timer of 10 minutes.
  4. Solve the matching SSC mensuration practice paper to test yourself in exam style.
  5. Check the previous year papers to see how SSC frames this topic.

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

Frequently Asked Questions

What is the difference between volume and surface area?

Volume is the space occupied by a solid, while surface area is the total area of its outer surface.

What is the difference between CSA and TSA?

CSA (curved surface area) excludes the base and top, while TSA (total surface area) includes them.

What is the volume of a cone?

The volume of a cone is 1/3 x pi x r squared x h.

What is the surface area of a sphere?

The surface area of a sphere is 4 x pi x r squared.

What happens when the radius of a sphere is doubled?

The surface area becomes 4 times and the volume becomes 8 times.

How do I solve melting and recasting questions?

The volume stays the same, so equate the volume of the original solid to the total volume of the new solids.

How many questions come from mensuration 3D in SSC exams?

One or two questions in Tier 1 and more in Tier 2.

How should I practise mensuration 3D?

Memorise the table, solve the examples here and attempt the mensuration practice paper.

Previous chapter

Mensuration 2D Notes for SSC Exams

Next chapter

Triangle Geometry Notes for SSC Exams
About the Author

Rahul Kumar

SSC Exam Expert & Content Editor

Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.

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