In this guide
Three-dimensional mensuration looks intimidating because there are many formulas. But SSC CGL uses the same few ideas again and again: volume is conserved when a solid is melted and recast; surface area is what gets painted or covered; and when dimensions scale by a factor, areas scale by the square and volumes by the cube. Learn the formulas in a table, then practise these three ideas.
Core formulas
| Solid | Volume | Surface area |
|---|---|---|
| Cube (a) | a³ | 6a² (lateral 4a²); diagonal a√3 |
| Cuboid (l, b, h) | lbh | 2(lb + bh + hl); diagonal √(l² + b² + h²) |
| Cylinder (r, h) | πr²h | curved 2πrh; total 2πr(r + h) |
| Cone (r, h, slant l) | ⅓πr²h | curved πrl; total πr(l + r); l = √(r² + h²) |
| Sphere (r) | (4/3)πr³ | 4πr² |
| Hemisphere (r) | (2/3)πr³ | curved 2πr²; total 3πr² |
| Frustum (R, r, h) | ⅓πh(R² + r² + Rr) | curved π(R + r)l |
| Prism | base area × height | lateral: perimeter × height |
Melting and recasting
Volume stays the same.
Worked example: A metal sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. How many small spheres are made?
Number = (6/2)³ = 27.
Worked example: A cylinder of radius 3 cm and height 8 cm is melted into a cone of the same radius. Find the cone's height.
πr² × 8 = ⅓πr²h → h = 24 cm.
Hollow solids
A hollow cylinder with outer radius R, inner radius r and height h:
volume of material = πh(R² − r²).
Worked example: A pipe is 14 m long with outer and inner radii of 4 cm and 3 cm. Find the volume of metal. (π = 22/7)
(22/7) × 1,400 × (16 − 9) = (22/7) × 1,400 × 7 = 30,800 cm³.
Scaling
If every linear dimension is multiplied by k:
- surface area is multiplied by k²;
- volume is multiplied by k³.
Worked example: If the radius of a sphere is doubled, its volume becomes 8 times and its surface area 4 times.
If only some dimensions change, apply each factor separately: for a cone whose radius doubles and height halves, the volume changes by 2² × ½ = 2 times.
Surface area questions
Worked example: Find the cost of painting the curved surface of a cylindrical pillar of radius 0.35 m and height 10 m at ₹20/m². (π = 22/7)
Curved area = 2 × (22/7) × 0.35 × 10 = 22 m². Cost = ₹440.
Worked example: A cube of side 6 cm is cut into cubes of side 2 cm. Find the increase in total surface area.
Original = 6 × 36 = 216. There are 27 small cubes, each with area 24, giving 648. Increase = 432 cm².
Water and flow problems
Volume flowing per unit time = cross-sectional area × speed.
Worked example: Water flows through a pipe of radius 1 cm at 7 m/s into a cylindrical tank of radius 50 cm. How much does the water level rise in 10 minutes? (π = 22/7)
Flow per second = π × 1² × 700 cm³. In 600 seconds: π × 4,20,000.
Rise in the tank = π × 4,20,000 ÷ (π × 2,500) = 168 cm.
Common traps
| Trap | Correct approach |
|---|---|
| Using diameter as radius | Halve it |
| Mixing units in flow problems | Convert m/s and m to cm |
| Using total surface for curved | Read what is painted or covered |
| Forgetting the slant height for a cone's surface | l = √(r² + h²) |
Practice
- Find the volume of a cube whose total surface area is 150 cm².
- Find the diagonal of a cuboid measuring 12 cm × 4 cm × 3 cm.
- A cone has radius 6 cm and height 8 cm. Find its curved surface area in terms of π.
- How many spheres of radius 1 cm can be made from a sphere of radius 5 cm?
- If the radius of a cylinder is halved and its height doubled, what happens to its volume?
- A hemisphere has radius 21 cm. Find its total surface area. (π = 22/7)
Answers: 1. 125 cm³. 2. 13 cm. 3. 60π cm². 4. 125. 5. It halves. 6. 4,158 cm².
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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