आयतन और पृष्ठीय क्षेत्रफल — त्रिविमीय क्षेत्रमिति
Volume and Surface Area — 3D Mensuration
Three-dimensional mensuration extends the same idea into solids: instead of area and perimeter you now find volume and surface area. SBI Clerk keeps this practical and formula-driven, sticking to five solids you can picture from daily life — cube, cuboid, cylinder, cone and sphere. Every question is again pure substitution once the right formula is in place, so the entire topic is a memory exercise rewarded with fast, certain marks. Learn the table, keep the units straight, and these questions fall in seconds.
The Five Solids and Their Formulas
Volume is measured in cubic units and surface area in square units, and keeping that distinction clear is half the battle. Total surface area counts every face, while curved or lateral surface area ignores the flat top and bottom — a distinction SBI Clerk loves to test. The table below carries the essential formulas; commit them to memory exactly, because a single misremembered term turns a 20-second question into a wrong answer with negative marking attached.
| Solid | Volume | Total Surface Area |
|---|---|---|
| Cube | a³ | 6a² |
| Cuboid | l × b × h | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πr(r + h) |
| Cone | ⅓πr²h | πr(r + l) |
| Sphere | 4/3 πr³ | 4πr² |
Cube and Cuboid
The cube depends on one length only: volume is the side cubed and total surface area is six times the side squared, since all six faces are identical. The cuboid needs length, breadth and height — volume is their product and surface area is twice the sum of the three face products. A frequent SBI Clerk twist gives the volume and two dimensions and asks for the third, which is just division. Keep the diagonal handy too: for a cuboid it is the square root of the sum of the squares of the three edges.
Cylinder, Cone and Sphere
The cylinder and cone share the base circle of radius r, so both use pi r squared, but the cone holds exactly one-third the volume of a cylinder of the same base and height. The cone's slant height l links to r and h by the Pythagoras relation, and you need l for its curved surface. The sphere depends only on its radius: volume is four-thirds pi r cubed and surface area is four pi r squared. A hemisphere is simply half the sphere's volume with a curved surface of two pi r squared.
Cone slant height: l = √(r² + h²) Cylinder curved SA = 2πrh Sphere volume = 4/3 πr³
Solved Examples
Example 1: A cylinder has radius 7 cm and height 10 cm. Volume = 22/7 × 49 × 10 = 1,540 cm³. Example 2: A cube has volume 343 cm³, so its side is the cube root, 7 cm, and its total surface area = 6 × 49 = 294 cm² — both answers from the single side length.
✗ Using total surface area when only the curved surface is asked | ✓ Reading whether the top and bottom faces are included
Melting, Recasting and Unit Traps
The favourite advanced SBI Clerk twist is melting one solid and recasting it as another — a sphere melted into cylinders, or a cuboid recast into cubes. The key insight is that volume is conserved: equate the two volumes and solve for the unknown, ignoring surface area entirely. The second trap is units: if a tank's dimensions are in metres but the answer wants litres, remember one cubic metre equals one thousand litres, and if lengths are mixed in metres and centimetres, convert everything to one unit before multiplying. These two ideas — conserve volume when reshaping, and standardise units before calculating — cover the handful of genuinely tricky 3D questions the exam sets, while the rest remain simple substitution into the formulas above.
आयतन घन इकाई में, पृष्ठीय क्षेत्रफल वर्ग इकाई में। कुल पृष्ठ में सभी फलक, वक्र पृष्ठ में ऊपर-नीचे नहीं। पिघलाकर ढालने पर आयतन समान रहता है।
Exam Pointer — 3D mensuration shows up in about 1–2 SBI Clerk Prelims questions, usually a direct volume or surface-area calculation on a cylinder, cube or sphere. Occasionally a melting-and-recasting question appears; solve it by equating volumes. Memorise the five formulas and you bank these marks in under 30 seconds each without any real reasoning.
60-Second Recap
- Volume in cubic units, surface area in square units.
- Cone volume is one-third of the cylinder with same base and height.
- Sphere: 4/3 πr³ volume, 4πr² surface — radius only.
- Melting problems conserve volume; always standardise units first.