Mensuration – An Exhaustive Study Guide (IIT‑JEE / Competitive Exams)
Mensuration is the branch of mathematics that deals with the measurement of lengths, areas, and volumes of geometric figures.
In competitive exams you will be tested on derivations, applications, and tricky geometry problems.
This guide covers the most important 2‑D and 3‑D shapes, key formulas, derivations, special techniques, and a hard practice problem.
1. 2‑D Shapes
| Shape | Area | Perimeter | Notes |
|---|---|---|---|
| Rectangle / Parallelogram | (A = b,h) | (P = 2(a+b)) | (b) = base, (h) = height |
| Square | (A = a^2) | (P = 4a) | |
| Triangle | (A = \frac{1}{2} b,h) | (P = a+b+c) | (h) = altitude to side (b) |
| Equilateral Triangle | (A = \frac{\sqrt{3}}{4} a^2) | (P = 3a) | |
| Isosceles Triangle | (A = \frac{1}{2} a,h) | (P = 2a + b) | |
| Scalene Triangle | Heron’s Formula: (A = \sqrt{s(s-a)(s-b)(s-c)}) | (P = a+b+c) | (s=\frac{a+b+c}{2}) |
| Circle | (A = \pi r^2) | (C = 2\pi r) | |
| Ellipse | (A = \pi a b) | (C \approx \pi[3(a+b)-\sqrt{(3a+b)(a+3b)}]) | (a,b) = semi‑axes |
| Regular Polygon | (A = \frac{n,s^2}{4\tan(\pi/n)}) | (P = n,s) | (n) = # sides, (s) = side length |
| Irregular Polygon | Shoelace Formula | Sum of side lengths |
1.1 Derivations
Triangle Area
- Using base & height: Drop a perpendicular from a vertex to the opposite side – the base of the triangle. The area of the rectangle formed by that base and its height is (b,h). Divide by 2 because a triangle is half a rectangle.
Circle Area (from integration)
[ A = \int_{-r}^{r} ! \sqrt{r^{2}-x^{2}};dx = 2\int_{0}^{r} \sqrt{r^{2}-x^{2}};dx ] Let (x = r\sin