Mensuration – Study Notes (IIT‑JEE / Competitive Exam Focus)
Program: NABARD Development Assistant
Subject: Quantitative Aptitude
Topic: Mensuration
Mensuration – the branch of Geometry that deals with the measurement of lengths, areas and volumes – is a core component of many competitive exams. In the following notes you will find:
- Rigorous derivations of every key formula.
- Exam‑oriented explanations highlighting the most common pitfalls.
- A concise formula sheet for quick reference.
- A hard practice problem with a detailed solution.
Tip: For every shape, write down the definition first (e.g., surface area of a sphere = area of its surface), then derive the formula. This structure helps you remember the logic behind the numbers rather than memorizing them blindly.
1. 2‑D Mensuration – Area
| Shape | Formula | Derivation / Key Idea | Notes |
|---|---|---|---|
| Rectangle | (A = l \times w) | Rectangle is a product of two perpendicular sides. | |
| Parallelogram | (A = b \times h) | Height is perpendicular to base. | If given an acute angle, use (b,h = \frac{1}{2}ab\sin\theta). |
| Triangle | (A = \frac{1}{2}ab\sin C) | Derived from base × height / 2; height = (b\sin C). | For right triangle, (\sin 90^\circ = 1). |
| Circle | (A = \pi r^2) | Limit of (n)-gon inscribed in circle → (\pi r^2). | |
| Ellipse | (A = \pi a b) | Stretch a circle of radius (r) by factors (a/r) and (b/r). | |
| Regular Polygon | (A = \frac{1}{2} P, r_g) (where (P) = perimeter, (r_g)= apothem) | Divide into (n) isosceles triangles. | |
| Quadrilateral (any) | (A = \sqrt{(s-a)(s-b)(s-c)(s-d)}) (Brahmagupta) | Generalization of Heron’s formula for cyclic quadrilaterals. | Only valid for cyclic quadrilaterals. |
| Triangle – Heron | (A = \sqrt{s(s-a)(s-b)(s-c)}) | (s = \frac{a+b+c}{2}). | Good for side‑only data. |
Derivation Highlights
-
Triangle area via vectors
For triangle with vertices (A(0,0)), (B(a,0)), (C(x_1,y_1)): [ A = \frac{1}{2}\left| \det\begin{pmatrix} a & 0 \ x_1 & y_1 \end{pmatrix}\right| = \frac{1}{2} |a y_1| = \frac{1}{2} b h. ] -
Circle area via polar coordinates
[ A = \int_0^{2\pi}\int_0^r r,dr,d\theta = \pi r^2. ]
2. 3‑D Mensuration – Volume & Surface Area
| Solid | Volume | Lateral Surface Area | Total Surface Area | Derivation / Key Idea |
|---|---|---|---|---|
| Cube | (V = a^3) | (6a^2) | (6a^2) | Simple product of side lengths. |
| Rectangular Prism | (V = lwh) | (2(lw+wh+lh)) | Same as lateral | |
| Cylinder | (V = \pi r^2 h) | (2 |