Introduction
3D Mensuration extends the same area/perimeter logic into volume and surface area — the "melting and
recasting" pattern (conserving volume while changing shape) is SSC's favourite twist on this chapter.
Volume & Surface Area Formula Reference
| Shape | Volume | Total Surface Area |
|---|
| Cube (side a) | a³ | 6a² |
| Cuboid (l, b, h) | l × b × h | 2(lb + bh + hl) |
| Cylinder (radius r, height h) | πr²h | 2πr(h+r) |
| Cone (radius r, height h, slant l) | ⅓ πr²h | πr(l+r), where l=√(r²+h²) |
| Sphere (radius r) | ⁴⁄₃ πr³ | 4πr² |
| Hemisphere (radius r) | ⅔ πr³ | 3πr² (total, incl. base) |
Q. Find the volume of a cylinder with radius 7 cm and height 10 cm.
Volume = πr²h = (22/7) × 7² × 10 = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 cm³
Melting & Recasting — Volume Conservation
Flowchart — Core Principle
When a solid is melted and recast into another shape, VOLUME stays the same
↓
Set Volume(original shape) = Volume(new shape), then solve for the unknown dimension
Q. A metallic sphere of radius 6 cm is melted and recast into small cones each of radius 3 cm and height 4 cm. Find the number of cones formed.
Volume of sphere = ⁴⁄₃ π(6)³ = ⁴⁄₃ π × 216 = 288π
Volume of one cone = ⅓ π(3)²(4) = ⅓ π × 36 = 12π
Number of cones = 288π / 12π = 24
Curved Surface Area vs Total Surface Area — Common Trap
⚠️ Common Trap: Always check whether a question asks for Curved Surface Area (CSA) (just the "wrap-around" part) or Total Surface Area (TSA) (CSA + base(s)). For a cylinder: CSA = 2πrh, TSA = 2πr(h+r). For a cone: CSA = πrl, TSA = πr(l+r).
Water/Tank Volume Word Problems
Q. A cylindrical tank of radius 7 m and height 10 m is full of water. Find how many litres of water it holds (1 m³ = 1000 litres).
Volume = πr²h = (22/7) × 49 × 10 = 1540 m³
In litres = 1540 × 1000 = 15,40,000 litres
✅ Practice Focus: Full volume/surface area table memorised · Melting-recasting volume conservation principle · CSA vs TSA distinction for cylinders and cones · Unit conversion for water/capacity problems (m³ ↔ litres).