📚 Punjab PSC – Quantitative Aptitude
📐 Topic: Mensuration
Mensuration is the branch of mathematics dealing with the measurement of length, area, perimeter, volume, and surface area of various geometric figures.
In the Punjab PSC exam, you will encounter a variety of problems that require quick recall of formulas and the ability to manipulate them efficiently.
📖 1. Simple Explanation
- Perimeter – the sum of all side lengths of a 2‑D figure.
- Area – the amount of 2‑D space inside a figure.
- Volume – the amount of 3‑D space a solid occupies.
- Surface Area (SA) – total area of all the outer faces of a 3‑D solid.
For each shape, memorise the basic formulas and know the trick to apply them quickly.
📌 2. Key Points & Formulas
| Figure | Perimeter / Circumference | Area | Volume | Surface Area |
|---|---|---|---|---|
| Square (side = a) | (4a) | (a^2) | – | – |
| Rectangle (length l, breadth b) | (2(l+b)) | (l\times b) | – | – |
| Rhombus (diagonals d1, d2) | – | (\dfrac{d_1d_2}{2}) | – | – |
| Parallelogram (base b, height h) | – | (b\times h) | – | – |
| Triangle (base b, height h) | – | (\dfrac{bh}{2}) | – | – |
| Isosceles Triangle (base b, sides l) | – | (\dfrac{b}{4}\sqrt{4l^2-b^2}) | – | – |
| Equilateral Triangle (side a) | – | (\dfrac{\sqrt3}{4}a^2) | – | – |
| Circle (radius r) | (\pi r) | (\pi r^2) | – | – |
| Trapezium (parallel sides a, b, height h) | – | (\dfrac{(a+b)h}{2}) | – | – |
| Cube (edge a) | – | – | (a^3) | (6a^2) |
| Cuboid (l, b, h) | – | – | (lbh) | (2(lb+bh+hl)) |
| Sphere (radius r) | – | – | (\dfrac{4}{3}\pi r^3) | (4\pi r^2) |
| Cylinder (radius r, height h) | – | – | (\pi r^2 h) | (2\pi r(r+h)) |
| Cone (radius r, slant height l, height h) | – | – | (\dfrac{1}{3}\pi r^2 h) | (\pi r(r+l)) |
| Right Circular Pyramid (base area B, slant height l, height h) | – | – | (\dfrac{1}{3}Bh) | (B + \dfrac{1}{2}\times \text{perimeter(base)} \times l) |
Quick Tips
- Remember π ≈ 22/7 – handy in exams where π is needed as a fraction.
- Area of Triangle via Heron’s Formula:
[ s = \frac{a+b+c}{2},\quad A=\sqrt{s(s-a)(s-b)(s-c)} ] Use only if the sides are given and not the height. - Triangular Base in Pyramids → use the area formula from 2‑D section.
- Cones & Cylinders share the same base area; volume is the base area × height × (1/3 for cone, 1 for cylinder).
- Surface Area of Cylinder: 2πr² (top & bottom) + 2πrh (lateral).
- Sphere: Volume = (4/3) × base area × radius.
🧮 3. Worked Example
Question (Typical PSC Style)
A right circular cone has a radius of 6 cm and a slant height of 10 cm. Find
- the height of the cone
- its volume
- its total surface area
Solution
-
Height
Use Pythagoras in the right triangle formed by radius, height, and slant height.
[ l^2 = r^2 + h^2 ;\Rightarrow; 10^2 = 6^2 + h^2 ;\Rightarrow; 100 = 36 + h^2 ;\Rightarrow; h^2 = 64 ;\Rightarrow; h = 8 \text{ cm} ] -
Volume
[ V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6)^2(8) = \frac{1}{3}\pi \times 36 \times 8 = 96\pi \text{ cm}^3 ] (If π = 22/7, then (V = 96 \times \frac{22}{7} = 302.4) cm³.) -
Surface Area
[ SA = \pi r(r + l) = \pi \times