What the exam asks
SBI Clerk focuses on 3D solids (cylinder, cone, sphere, hemisphere, cuboid) plus basic 2D figures. Use pi = 22/7 unless told otherwise; choose radii that are multiples of 7 for clean answers. Target: 45 seconds.
3D formulas
| Solid | Curved/lateral SA | Total SA | Volume |
|---|---|---|---|
| Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | lbh |
| Cube (a) | 4a² | 6a² | a³ |
| Cylinder (r, h) | 2 pi r h | 2 pi r (r + h) | pi r² h |
| Cone (r, h, l) | pi r l | pi r (l + r) | (1/3) pi r² h |
| Sphere (r) | 4 pi r² | 4 pi r² | (4/3) pi r³ |
| Hemisphere (r) | 2 pi r² | 3 pi r² | (2/3) pi r³ |
Slant height of a cone: l = sqrt(r² + h²). Useful triples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17).
2D formulas
- Rectangle: area lb, perimeter 2(l + b), diagonal sqrt(l² + b²).
- Square: area a², diagonal a sqrt(2).
- Circle: area pi r², circumference 2 pi r. Semicircle perimeter = pi r + 2r.
- Triangle: (1/2) x base x height; equilateral: (sqrt(3)/4) a².
- Path of width w outside a rectangle: 2w(l + b + 2w); inside: 2w(l + b - 2w).
Melting and recasting
Volume is conserved: number of small solids = volume of big solid / volume of one small solid. pi cancels, so you rarely multiply by 22/7.
Percentage change in volume
- Cylinder: r up 20%, h down 20% gives 1.2² x 0.8 = 1.152, a 15.2% increase.
- Sphere: r up 10% gives volume up (1.1)³ - 1 = 33.1%, surface area up 21%.
Worked example
Cone with r = 7 cm and h = 24 cm: l = 25. CSA = 22/7 x 7 x 25 = 550 sq cm; volume = (1/3) x 22/7 x 49 x 24 = 1,232 cu cm.
Worked example 2
Sphere of radius 21 cm: surface area = 4 x 22/7 x 441 = 5,544 sq cm; volume = (4/3) x 22/7 x 9,261 = 38,808 cu cm.
Common traps
- Using diameter as radius.
- Curved vs total surface area (an open cylinder has only one base).
- Units: 1 litre = 1,000 cu cm; 1 cu m = 1,000 litres.
- Hemisphere TSA is 3 pi r², not 2 pi r².
Worked example 3 (cylinder)
For a closed cylinder with r = 7 cm and h = 10 cm: CSA = 2 x 22/7 x 7 x 10 = 440 sq cm, TSA = 440 + 2 x 154 = 748 sq cm, volume = 22/7 x 49 x 10 = 1,540 cu cm.