Mensuration – Advanced Study Notes
(Quantitative Aptitude – Madhya Pradesh TET – University‐Level Treatment)
1. Overview
Mensuration is the branch of geometry dealing with the measurement of lengths, areas, and volumes of geometric figures.
In an aptitude exam you will encounter:
| 2‑D Figures | 3‑D Figures |
|---|---|
| Triangle, Quadrilateral, Polygon, Circle, Ellipse | Cube, Cuboid, Sphere, Cylinder, Cone, Prism, Pyramid, Torus |
| Solids of revolution (cylindrical & spherical) |
The key to success is understanding the first‑principle derivation of the formulas, the assumptions involved, and the edge cases that can trip you up.
2. Fundamental Concepts
| Concept | Definition | Why It Matters |
|---|---|---|
| Coordinate Geometry | Representation of points as ((x,y)) or ((x,y,z)) | Enables analytic derivation of areas/volumes |
| Pythagorean Theorem | (a^2+b^2=c^2) for right triangles | Foundation for distance, radius, and diagonal calculations |
| Trigonometry | ( \sin, \cos, \tan) relations in a triangle | Required for non‑right triangles, sector area, etc. |
| Integration (Calculus) | (\int f(x),dx) | Needed for volumes by revolution, surface area, and irregular shapes |
| Symmetry and Decomposition | Breaking complex figures into simpler parts | Simplifies calculations and reveals hidden patterns |
3. Two‑Dimensional Mensuration
3.1 Polygons
| Polygon | Area |
|---|---|
| Triangle | (\displaystyle \frac{1}{2},b,h) |
| Quadrilateral (any) | (\displaystyle \frac{1}{2},d_1d_2\sin\theta) where (d_1,d_2) are diagonals, (\theta) angle between them |
| Regular (n)-gon | (\displaystyle \frac{n,s^2}{4\tan(\pi/n)}) where (s) is side length |
Derivation (Regular (n)-gon)
Divide into (n) isosceles triangles from the centre.
Area of one = (\frac{1}{2} s \cdot r) where (r) is apothem.
Using (s = 2r \tan(\pi/n)), substitute and sum over (n).
3.2 Circles & Ellipses
| Figure | Area | Circumference |
|---|---|---|
| Circle | (\displaystyle \pi r^2) | (\displaystyle 2\pi r) |
| Ellipse | (\displaystyle \pi ab) | (4E(e)) (elliptic integral) – approximate: (\displaystyle \pi\bigl[3(a+b)-\sqrt{(3a+b)(a+3b)}\bigr]) |
Edge Cases
- Degenerate ellipse: (a=b) → circle.
- Zero radius: area = 0.
- Negative radius: Not physically meaningful; treat as 0.
3.3 Triangular Areas via Trigonometry
| Formula | Conditions |
|---|---|
| (A = \frac{1}{2}ab\sin C) | (a,b) are sides, (C) the included angle |
| Heron’s Formula | (s=\frac{a+b+c}{2}); (A=\sqrt{s(s-a)(s-b)(s-c)}) |
Proof (Heron)
Construct altitude (h) from side (c). Express (h) in terms of sides using Pythagoras, eliminate (h) to obtain the expression.
4. Three‑Dimensional Mensuration
4.1 Primitives
| Figure | Volume |
|---|---|
| Cube | (s^3) |
| Cuboid | (lwh) |
| Cylinder | (\displaystyle \pi r^2 h) |
| Cone | (\displaystyle \frac{1}{3}\pi r^2 h) |
| Sphere | (\displaystyle \frac{4}{3}\pi r^3) |
| Pyramid | (\displaystyle \frac{1}{3}Bh) (B = base area) |
Derivation (Sphere)
Using calculus:
[
V = \int_{-r}^{r} \pi (r^2 - y^2),dy = \frac{4}{3}\pi r^3
]
4.2 Solids of Revolution
| Solid | Formula | Derivation |
|---|---|---|
| Cylinder (via disc method) | (\displaystyle \pi r^2 h) | (\int_0^h \pi r^2,dz) |
| Cone (via disc method) | (\displaystyle \frac{1}{3}\pi r^2 h) | (\int_0^h \pi \left(\frac{r}{h}z\right)^2,dz) |
| Torus | (\displaystyle V = 2\pi^2 R r^2) | Revolve a circle radius (r) |