Mensuration means measuring the size of shapes: the length around them (perimeter), the space they cover (area) and the space they fill (volume). Lok Sewa quantitative questions use a small set of formulas again and again. This note collects them in clear tables, adds the Nepali land units that are often asked, and shows worked examples so you can solve each type quickly.
1. Perimeter and area of flat shapes
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle (length l, breadth b) | 2(l + b) | l × b |
| Square (side a) | 4a | a² |
| Triangle (base b, height h) | sum of the three sides | ½ × b × h |
| Right-angled triangle (legs a and b) | a + b + hypotenuse | ½ × a × b |
| Circle (radius r) | 2πr (circumference) | πr² |
| Parallelogram (base b, height h) | 2 × (sum of two adjacent sides) | b × h |
In exams π is usually taken as 22/7. Choose a radius that is a multiple of 7 when you can, because the 7 cancels neatly.
2. Worked examples: flat shapes
- A rectangle is 12 m long and 8 m wide. Area = 12 × 8 = 96 m²; perimeter = 2(12 + 8) = 40 m.
- A square has a perimeter of 48 cm, so each side is 48 ÷ 4 = 12 cm and the area is 12² = 144 cm².
- A circle has radius 7 cm. Area = 22/7 × 7 × 7 = 154 cm²; circumference = 2 × 22/7 × 7 = 44 cm.
- A circle has a circumference of 88 cm. Then 2 × 22/7 × r = 88, so r = 88 × 7 ÷ 44 = 14 cm.
- A triangle has base 10 cm and height 6 cm. Area = ½ × 10 × 6 = 30 cm².
3. The Pythagoras rule
In a right-angled triangle, hypotenuse² = base² + height². Learn the common sets of whole-number sides: 3, 4, 5; 6, 8, 10; 5, 12, 13; 8, 15, 17. For example, legs of 6 cm and 8 cm give a hypotenuse of 10 cm, and the area is ½ × 6 × 8 = 24 cm².
4. Volume and surface area of solids
| Solid | Volume | Surface area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid (l, b, h) | l × b × h | 2(lb + bh + hl) |
| Cylinder (radius r, height h) | πr²h | curved: 2πrh; total: 2πr(r + h) |
| Cone (radius r, height h) | ⅓ πr²h | curved: πrl (l = slant height) |
| Sphere (radius r) | ⁴⁄₃ πr³ | 4πr² |
| Hemisphere (radius r) | ⅔ πr³ | total: 3πr² |
5. Worked examples: solids
- A cube has side 5 cm. Volume = 5³ = 125 cm³; total surface area = 6 × 25 = 150 cm².
- A box is 6 cm × 4 cm × 3 cm. Volume = 6 × 4 × 3 = 72 cm³.
- A cylinder has radius 7 cm and height 10 cm. Volume = 22/7 × 49 × 10 = 1540 cm³; curved surface = 2 × 22/7 × 7 × 10 = 440 cm².
- The four walls of a room 5 m long, 4 m wide and 3 m high have area 2 × height × (l + b) = 2 × 3 × 9 = 54 m².
6. Units and conversions
| Conversion | Value |
|---|---|
| 1 m | 100 cm |
| 1 km | 1000 m |
| 1 m² | 10,000 cm² |
| 1 hectare | 10,000 m² |
| 1 m³ | 1000 litres |
| 1 litre | 1000 cm³ (1000 ml) |
7. Nepali land units
Land in Nepal is measured in two systems: ropani–aana–paisa–daam in the hills and bigha–kattha–dhur in the Terai.
| Hill system | Equal to | Terai system | Equal to |
|---|---|---|---|
| 1 ropani | 16 aana (5476 sq ft, about 508.72 m²) | 1 bigha | 20 kattha (72,900 sq ft, about 6772.63 m²) |
| 1 aana | 4 paisa (342.25 sq ft) | 1 kattha | 20 dhur (3645 sq ft, about 338.63 m²) |
| 1 paisa | 4 daam | 1 dhur | 182.25 sq ft |
8. How changes in size affect results
- If the side of a square is doubled, the perimeter doubles but the area becomes 4 times.
- If the radius of a circle is tripled, the area becomes 9 times.
- If the side of a cube is doubled, the volume becomes 8 times.
- In general, area changes with the square of the change in length, and volume with the cube.
📌 Key points to remember
- Rectangle: area l × b, perimeter 2(l + b).
- Circle: area πr², circumference 2πr; take π = 22/7.
- Triangle area = ½ × base × height.
- Pythagoras: 3-4-5, 6-8-10, 5-12-13, 8-15-17.
- Cube: volume a³, surface 6a²; cylinder: volume πr²h.
- Four walls = 2 × height × (length + breadth).
- 1 m³ = 1000 litres; 1 hectare = 10,000 m².
- 1 ropani = 16 aana; 1 bigha = 20 kattha; 1 kattha = 20 dhur.
Test yourself: Mensuration
Practise Perimeter and area of flat shapes, Worked examples: solids and How changes in size affect results with 20 questions.