Mensuration – A Concise Study Guide for UPSC EPFO
Mensuration is the branch of mathematics that deals with the measurement of lengths, areas, perimeters, surface areas, and volumes of geometric figures.
The EPFO exam often tests basic formulas and quick problem‑solving skills, so this guide focuses on the essential concepts and handy shortcuts.
1. Simple Explanation
- Length – distance between two points (e.g., side of a rectangle).
- Perimeter – total length of the boundary of a 2‑D figure.
- Area – amount of flat surface inside the boundary.
- Surface Area (SA) – total area of all faces of a 3‑D figure.
- Volume (V) – amount of space occupied by a 3‑D figure.
Tip: Always keep units in mind (cm, m, km, etc.). In exam questions, if units aren’t specified, you can work in any consistent unit and just give the numerical value.
2. Key Points & Formulas
| Figure | Perimeter / Surface Area | Area / Volume | Notes |
|---|---|---|---|
| Rectangle / Square | (P = 2(l + w)) | (A = l \times w) | For a square, (l = w = a). |
| Parallelogram | (P = 2(a + b)) | (A = \text{base} \times \text{height}) | Height is perpendicular to base. |
| Rhombus | (P = 4a) | (A = \frac{d_1 \times d_2}{2}) | (d_1,d_2) are diagonals. |
| Trapezium (trapezoid) | (P = a + b + c + d) | (A = \frac{(a + c) \times h}{2}) | (a,c) are parallel sides; (h) is perpendicular height. |
| Circle | (C = 2\pi r) | (A = \pi r^2) | (r) = radius. |
| Triangle | (P = a + b + c) | (A = \frac{1}{2} \times \text{base} \times \text{height}) | Use Heron’s formula if only sides known: (A=\sqrt{s(s-a)(s-b)(s-c)}), (s=\frac{a+b+c}{2}). |
| Sphere | (SA = 4\pi r^2) | (V = \frac{4}{3}\pi r^3) | |
| Cylinder | (SA = 2\pi r(r+h)) | (V = \pi r^2 h) | (h)=height. |
| Cone | (SA = \pi r(r + l)) | (V = \frac{1}{3}\pi r^2 h) | (l)=slant height, (h)=vertical height. |
| Right Circular Cone | Same as cone | Same as cone | |
| Cuboid (rectangular prism) | (SA = 2(lw + lh + wh)) | (V = l \times w \times h) | |
| Cube | (SA = 6a^2) | (V = a^3) | |
| Regular Pyramid | (SA = B + \frac{1}{2} \times P \times \ell) | (V = \frac{1}{3} B h) | (B)=base area, (P)=perimeter of base, (\ell)=slant height, (h)=vertical height. |
| Regular Prism | (SA = 2B + P \times h) | (V = B \times h) | (B)=area of base, (P)=perimeter of base. |
Common abbreviations:
(P) – perimeter, (A) – area, (V) – volume, (SA) – surface area, (r) – radius, (h) – height, (l) – slant height, (s) – semiperimeter.
3. Worked Example
Problem
A right circular cone has a radius of (6\ \text{cm}) and a height of (8\ \text{cm}).
(a) Find the slant height of the cone.
(b) Compute the volume.
(c) Find the total surface area.
Solution
(a) Slant height ((l))
Use the Pythagorean theorem in the cross‑section triangle:
(l^2 = r^2 + h^2)
(l^2 = 6^2 + 8^2 = 36 + 64 = 100)
(l = \sqrt{100} = 10\ \text{cm})
(b) Volume ((V))
(V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6)^2 (8))