Mensuration – Tripura PSC Quantitative Aptitude
(For the 2026 PSC Examination – Quantitative Aptitude Section)
1. Simple Explanation
Mensuration is the branch of mathematics that deals with finding the area, perimeter, volume, and surface area of various plane and solid figures.
- Plane figures – shapes that lie in a single 2‑D plane (e.g., rectangle, triangle, circle).
- Solid figures – 3‑D shapes that occupy space (e.g., cube, cylinder, cone).
The key is to remember the basic formulae and the relationships between the different dimensions of a figure.
2. Key Points & Formulae
| Figure | Symbol | Perimeter | Area | Volume | Surface Area |
|---|---|---|---|---|---|
| Rectangle / Square | – | (P = 2(l+b)) | (A = l \times b) | – | – |
| Parallelogram | – | (P = 2(a+b)) | (A = \text{base} \times \text{height}) | – | – |
| Rhombus | – | (P = 4a) | (A = \frac{d_1 \times d_2}{2}) | – | – |
| Triangle | – | (P = a+b+c) | (A = \frac{1}{2} \times \text{base} \times \text{height}) | – | – |
| Right‑Angled Triangle | – | – | (A = \frac{1}{2}ab) (with legs (a,b)) | – | – |
| Circle | – | (C = 2\pi r) | (A = \pi r^2) | – | – |
| Ellipse | – | – | (A = \pi a b) | – | – |
| Trapezium (Trapezoid) | – | – | (A = \frac{1}{2}(a+b)\times h) | – | – |
| Cube | – | (P = 12a) | (A = 6a^2) | (V = a^3) | – |
| Cuboid (Rectangular Prism) | – | – | (A = 2(lw+lh+wh)) | (V = lwh) | – |
| Sphere | – | – | – | (V = \frac{4}{3}\pi r^3) | (S = 4\pi r^2) |
| Cylinder | – | – | – | (V = \pi r^2 h) | (S = 2\pi r(h+r)) |
| Cone | – | – | – | (V = \frac{1}{3}\pi r^2 h) | (S = \pi r(r+s)) (where (s) is slant height) |
| Pyramid (regular) | – | – | – | (V = \frac{1}{3} B h) | (S = B + \frac{1}{2} P l) (where (P) is base perimeter, (l) is slant height) |
| Prism (regular) | – | – | – | (V = B \times h) | (S = 2B + Ph) |
Important Constants
| Symbol | Value | Notes |
|---|---|---|
| (\pi) | 3.1416 (≈ 22/7) | Use 22/7 for quick mental calculations |
| ( \sqrt{3} ) | 1.732 | Often appears in equilateral triangles |
Quick Tips
- Unit Consistency – Always keep units consistent (cm, m, etc.) before plugging into formulas.
- Right‑Triangle Relations – Use Pythagoras for slant heights: (s = \sqrt{r^2 + h^2}).
- Area of Triangle from Coordinates – Use determinant formula if points are given.
- Surface Area of Cylinder – Remember to add top and bottom circles (the first term in (S)).
- Volume of Pyramid vs. Prism – Both have (V = \text{Base area} \times \text{height}); pyramid has an extra factor (1/3).
3. Worked Example
Question
A right circular cylinder has a radius of 5 cm and a height of 12 cm.
(a) Find its volume.
(b) Find its total surface area.
Solution
Given
- Radius (r = 5) cm
- Height (h = 12) cm
(a) Volume
[ V = \pi r^2 h = \pi (5)^2 (12) = \pi \times 25 \times 12 = 300\pi \text{ cm}^3 ]
Using (\pi \approx 22/7):
[ V \approx 300 \times \frac{22}{7} = \frac{6600}{7} \approx 942.86 \text{ cm}^3 ]
(b) Total Surface Area
[ S = 2\pi r (h + r) = 2\pi (5) (12 + 5) = 10\pi \times 17 = 170\pi \text{ cm}^2 ]
Approximate:
[ S \approx 170 \times \frac{22}{7} = \frac{3740}{7} \approx 534.29 \text{ cm}^2 ]
Answer
- Volume ≈ **942.86