Mensuration – Goa PSC Quantitative Aptitude
(Study Notes for Quick Revision & Practice)
Mensuration refers to the branch of mathematics dealing with the measurement of geometric figures – mainly their area, perimeter, and volume.
For the Goa PSC, the focus is on 2‑D shapes (area/perimeter) and 3‑D solids (volume).
📌 Simple Explanation
| Concept | What It Means | How It’s Used |
|---|---|---|
| Area | The amount of space inside a figure (square units). | Finding the size of a field, floor, wall, etc. |
| Perimeter | The total length around a figure (linear units). | Calculating boundary length, fence, etc. |
| Volume | The amount of space occupied by a 3‑D object (cubic units). | Capacity of containers, storage space, etc. |
| Radius / Diameter / Radius of Curvature | Measures of circles. | Needed in all circle‑related problems. |
📑 Key Points & Formulas
| Shape | Area Formula | Perimeter / Circumference | Volume (if applicable) |
|---|---|---|---|
| Rectangle | (A = l \times w) | (P = 2(l + w)) | – |
| Square | (A = s^2) | (P = 4s) | – |
| Triangle | (A = \frac{1}{2} \times b \times h) | (P = a + b + c) | – |
| Parallelogram | (A = \text{base} \times \text{height}) | (P = 2(a + b)) | – |
| Trapezium / Trapezoid | (A = \frac{1}{2}(a + b) \times h) | (P = a + b + c + d) | – |
| Circle | (A = \pi r^2) | (C = 2\pi r = \pi d) | – |
| Sphere | – | – | (V = \frac{4}{3}\pi r^3) |
| Cylinder | – | – | (V = \pi r^2 h) |
| Cone | – | – | (V = \frac{1}{3}\pi r^2 h) |
| Cuboid | – | – | (V = l \times w \times h) |
| Cube | – | – | (V = s^3) |
| Prism | – | – | (V = \text{Area of base} \times \text{height}) |
| Pyramid | – | – | (V = \frac{1}{3}\times \text{Area of base} \times \text{height}) |
Common constants
- (\pi \approx 22/7) or (3.14) (use whichever gives a simpler calculation).
- For quick mental math, use (\pi = 22/7) to keep the result rational.
📐 Worked Example
Problem:
A circular pond has a radius of 12 cm.
(a) Find its area.
(b) If the pond is to be fenced, what is the minimum length of the fence?
(c) The pond’s depth is 3 cm. What is the volume of water it can hold?
Solution
-
Area of the pond
[ A = \pi r^2 = \frac{22}{7} \times 12^2 = \frac{22}{7} \times 144 ] [ = \frac{22 \times 144}{7} = \frac{3168}{7} \approx 452.57 \text{ cm}^2 ] -
Length of the fence (circumference)
[ C = 2\pi r = 2 \times \frac{22}{7} \times 12 = \frac{44}{7} \times 12 ] [ = \frac{528}{7} \approx 75.43 \text{ cm} ] -
Volume of water
The pond is a cylindrical shape (flat top, depth as height).
[ V = \pi r^2 h = \frac{22}{7} \times 12^2 \times 3 ] [ = \frac{22}{7} \times 144 \times 3 = \frac{22 \times 432}{7} ] [ = \frac{9504}{7} \approx 1357.71 \text{ cm}^3 ]
Answers:
- Area ≈ 452.57 cm²
- Fence length ≈ 75.43 cm
- Water volume ≈ 1357.71 cm³
Tip: For quick PSC style computations, keep the fraction form and only convert to decimal if required.
🔄 Quick Revision Summary
| Concept | Formula (short form) | Quick Tip |
|---|---|---|
| Rectangle | (A = l w) | Double the sum of sides for perimeter. |
| Square | (A = s^2) | Perimeter = (4s). |
| Triangle | (A = \frac{1}{2} b h) | Use base × height ÷ 2. |
| Parallelogram | (A = \text{base}\times\text{height}) | Same as rectangle but slanted. |
| Trapezium | (A |