Active and passive voice: rules for every tense
How the verb changes in each tense, pronoun changes, modals and imperative sentences, with SSC examples.
Study notes
Mensuration formula sheet: 2D and 3D shapes
Area, perimeter, volume and surface area for every shape SSC asks about, in one printable sheet.
Mock Matrix · mockmatrix.in/
2D shapes
| Shape | Area | Perimeter |
|---|---|---|
| Square (side a) | a² | 4a |
| Rectangle (l, b) | l × b | 2(l + b) |
| Triangle | ½ × base × height | sum of sides |
| Equilateral triangle (a) | (√3/4) a² | 3a |
| Circle (r) | πr² | 2πr |
| Semicircle (r) | πr²/2 | πr + 2r |
| Parallelogram | base × height | 2(a + b) |
| Rhombus | ½ × d₁ × d₂ | 4a |
| Trapezium | ½ × (sum of parallel sides) × height | sum of sides |
Heron’s formula: Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2
Diagonal of a square = a√2 · Diagonal of a rectangle = √(l² + b²)
Diagonal of a square = a√2 · Diagonal of a rectangle = √(l² + b²)
3D shapes
| Solid | Volume | Curved / lateral surface | Total surface |
|---|---|---|---|
| Cube (a) | a³ | 4a² | 6a² |
| Cuboid (l, b, h) | lbh | 2h(l + b) | 2(lb + bh + hl) |
| Cylinder (r, h) | πr²h | 2πrh | 2πr(r + h) |
| Cone (r, h, slant l) | ⅓πr²h | πrl | πr(l + r) |
| Sphere (r) | ⁴⁄₃πr³ | 4πr² | 4πr² |
| Hemisphere (r) | ⅔πr³ | 2πr² | 3πr² |
Slant height of a cone: l = √(r² + h²). Diagonal of a cuboid: √(l² + b² + h²).
Solved examples
- A circle has circumference 44 cm. Find its area (π = 22/7).
2πr = 44 → r = 7. Area = 22/7 × 49 = 154 cm². - Find the volume of a cone with radius 3 cm and height 7 cm.
⅓ × 22/7 × 9 × 7 = 66 cm³. - The side of a square is increased by 20%. By what percentage does its area increase?
1.2² = 1.44 → 44%.
Shortcut: if every length of a shape changes by r%, area changes by 2r + r²/100 % and volume by the three-step successive change.
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