Mensuration – Quantitative Aptitude
For the West Bengal Patwari / Lekhpal Examination
Mensuration deals with the measurement of the size, area, length, and volume of geometric figures.
1. Simple Explanation
- Perimeter – distance around a 2‑D figure.
- Area – amount of surface covered by a 2‑D figure.
- Surface Area – total area of all faces of a 3‑D figure.
- Volume – amount of space occupied by a 3‑D figure.
All these quantities are expressed in square units (e.g., m², cm²) for area and cubic units (e.g., m³, cm³) for volume.
2. Key Points & Formulas
| Figure | Perimeter / Lateral (if applicable) | Area | Surface Area | Volume |
|---|---|---|---|---|
| Square | (P = 4a) | (A = a^2) | — | — |
| Rectangle | (P = 2(l+w)) | (A = l \times w) | — | — |
| Parallelogram | — | (A = b \times h) | — | — |
| Rhombus | (P = 4a) | (A = \frac{d_1 d_2}{2}) | — | — |
| Trapezium | — | (A = \frac{(a+b)\times h}{2}) | — | — |
| Triangle | — | (A = \frac{b \times h}{2}) | — | — |
| Circle | (C = 2\pi r) | (A = \pi r^2) | — | — |
| Ellipse | — | (A = \pi a b) | — | — |
| Cube | — | — | (SA = 6a^2) | (V = a^3) |
| Cuboid | (L = 2(l+w+h)) | — | (SA = 2(lw+wh+hl)) | (V = l \times w \times h) |
| Sphere | — | — | (SA = 4\pi r^2) | (V = \frac{4}{3}\pi r^3) |
| Cylinder | (L = 2\pi r + 2\pi r) | — | (SA = 2\pi r (r + h)) | (V = \pi r^2 h) |
| Cone | (L = \pi r + \pi r \sqrt{r^2+h^2}) | — | (SA = \pi r (r + \sqrt{r^2+h^2})) | (V = \frac{1}{3}\pi r^2 h) |
| Pyramid | — | — | (SA = \text{Base area} + \frac{1}{2}\times \text{Perimeter of base} \times \text{slant height}) | (V = \frac{1}{3}\times \text{Base area} \times h) |
| Frustum of a Cone | — | — | (SA = \pi (r_1 + r_2)(\text{slant height}) + \pi r_1^2 + \pi r_2^2) | (V = \frac{1}{3}\pi h (r_1^2 + r_1 r_2 + r_2^2)) |
Conversion Notes
| Unit | Cubic Meter | Cubic Centimeter | Cubic Meter | Cubic Centimeter |
|---|---|---|---|---|
| 1 m³ | 1 | 1 000 000 | 1 | 1 000 000 |
| 1 cm³ | (1\times10^{-6}) | 1 | (1\times10^{-6}) | 1 |
- π ≈ 3.14 (use 22/7 for simple fractions).
- Use Pythagoras for slant heights: (l = \sqrt{r^2 + h^2}).
3. Worked Example
Problem
A trapezium has the following dimensions:
- Upper base (a = 8 \text{ m})
- Lower base (b = 14 \text{ m})
- Height (h = 6 \text{ m})
Find its area and perimeter if the non‑parallel sides are equal in length and each is (7 \text{ m}).
Solution
- Area
[ A = \frac{(a + b) \times h}{2} = \frac{(8 + 14) \times 6}{2} = \frac{22 \times 6}{2} = \frac{132}{2} = 66 \text{ m}^2 ]
- Perimeter
[ P = a + b + 2 \times \text{(non‑parallel side)} = 8 + 14 + 2 \times 7 = 22 + 14 = 36 \text{ m} ]
Answer:
- Area = 66 m²
- Perimeter = 36 m
Tip: For irregular shapes, split them into familiar shapes and add the areas.
4. Quick Revision Summary
- Perimeter: sum of all side lengths.
- Area:
- Rectangle: (l \times w)
- Triangle: (\frac{1}{2} \times b \times h\