Mensuration – Study Notes for Manipur PSC (Quantitative Aptitude)
Mensuration is the branch of mathematics dealing with the measurement of geometric figures.
It is crucial for the PSC exam because many questions ask you to find area, perimeter, volume, or surface area of various shapes.
1. Simple Explanation
- Area: Surface covered by a shape.
- Perimeter: Sum of all side lengths.
- Volume: Space occupied by a 3‑D object.
- Surface Area: Total area of all exposed surfaces of a 3‑D object.
All formulas are derived from basic geometry; memorize them and practice using the “units” (cm, m, etc.) to avoid errors.
2. Key Formulas & Concepts
| Shape | Area | Perimeter | Volume | Surface Area (if applicable) |
|---|---|---|---|---|
| Rectangle | (A = l \times w) | (P = 2(l+w)) | – | – |
| Square | (A = a^2) | (P = 4a) | – | – |
| Triangle | (A = \frac{1}{2} \times b \times h) Isosceles: (A = \frac{b^2}{4}\sqrt{4a^2-b^2}) |
(P = a+b+c) | – | – |
| Parallelogram | (A = b \times h) | (P = 2(a+b)) | – | – |
| Trapezium | (A = \frac{1}{2}(a+b)h) | (P = a+b+c+d) | – | – |
| Circle | (A = \pi r^2) | (C = 2\pi r) | – | – |
| Ellipse | (A = \pi a b) | – | – | – |
| Regular Polygon (n sides, side (s)) | (A = \frac{n s^2}{4\tan(\pi/n)}) | (P = n s) | – | – |
| Cube | – | – | (V = a^3) | (S = 6a^2) |
| Cuboid | – | – | (V = lwh) | (S = 2(lw+wh+hl)) |
| Cylinder | – | – | (V = \pi r^2 h) | (S = 2\pi r(r+h)) |
| Cone | – | – | (V = \frac{1}{3}\pi r^2 h) | (S = \pi r(r + l)) (l = slant height) |
| Sphere | – | – | (V = \frac{4}{3}\pi r^3) | (S = 4\pi r^2) |
Quick‑Tip for Trapezium
- If the trapezium is right‑angled and one side is parallel to the base, use (h) as the vertical height.
Common Mistakes
- Mixing up radius & diameter in circle formulas.
- Forgetting the (1/2) factor in triangle area.
- Using (4) instead of (2) for the perimeter of a rectangle.
- Converting units incorrectly (e.g., cm to m) when calculating volume.
3. Worked Example
Question:
A right‑angled triangular prism has a base triangle with sides 3 cm, 4 cm, and 5 cm. The height of the prism is 10 cm. Find the volume and the total surface area.
- Identify the base shape – Right‑angled triangle (3‑4‑5).
- Area of base
[ A_{\text{base}} = \frac{1}{2}\times3 \times 4 = 6 \text{ cm}^2 ] - Volume (Prism formula)
[ V = A_{\text{base}}\times h = 6 \times 10 = 60 \text{ cm}^3 ] - Surface area
- Two triangular faces: (2 \times 6 = 12)
- Lateral area (sum of rectangles connecting corresponding sides)
[ \text{Lateral} = h (a+b+c) = 10(3+4+5)=10\times12=120 ] - Total SA
[ S = 12 + 120 = 132 \text{ cm}^2 ] Answer: (V = 60\ \text{cm}^3,\ S = 132\ \text{cm}^2)
4. Quick Revision Summary
| Concept | Formula | Key Insight |
|---|---|---|
| Area of rectangle | (l \times w) | Product of length & width |
| Perimeter of rectangle | (2(l+w)) | Add length & width, double |
| Area of triangle | (\frac{1}{2} b h) | Base × Height ÷ 2 |
| Area of circle | (\pi r^2) | Radius squared × π |
| Volume of cylinder | (\pi r^2 h) | Base area × height |
| Volume of cone | (\frac{1}{3}\pi r^2 h) | One‑third of cylinder |
| Volume of sphere | (\frac{4}{3}\pi r^3) | 4/3 times radius cubed × π |
| Surface area of cube | (6a^2) | 6 faces of equal area |
| Surface area of sphere | (4\pi r^2) | 4π times radius squared |
- Remember: Always keep units consistent.