Mensuration – Advanced Study Notes
(Prepared for the “Meghalaya Patwari / Lekhpal” Quantitative Aptitude Course)
Goal – Equip you with a rigorous, first‑principles understanding of area, perimeter, volume and surface area calculations for a wide spectrum of geometrical figures.
Audience – Students preparing for competitive aptitude exams (Patwari, Lekhpal, and similar Indian civil‑service entrance tests).
Depth – MIT‑style: derivations, edge‑case analysis, and a challenging problem that tests both theory and intuition.
1. Foundations
1.1 Euclidean Plane Geometry
- Points, Lines, Planes – Defined by axioms (Euclid's postulates).
- Congruence & Similarity – Two figures are congruent if a sequence of rigid motions (translations, rotations, reflections) maps one onto the other.
- Angles – Measured in degrees or radians.
1.2 Coordinate Geometry
- Cartesian Coordinates – Every point (P(x, y)) in the plane.
- Distance Formula – (\displaystyle d(P,Q)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}).
- Slope – (m=\frac{y_2-y_1}{x_2-x_1}).
1.3 Basic Calculus (for area/volume derivation)
- Riemann Sums – Approximating area under a curve.
- Fundamental Theorem of Calculus – Links differentiation and integration.
2. Two‑Dimensional Figures
| Figure | Key Properties | General Formula | Derivation Sketch |
|---|---|---|---|
| Rectangle | Opposite sides equal, 4 right angles | (A = lw), (P = 2(l+w)) | Direct from definition |
| Parallelogram | Opposite sides parallel, equal | (A = bh), (P = 2(a+b)) | (b) is base, (h) height |
| Triangle | 3 sides, 3 angles | (A = \frac{1}{2}bh), (P = a+b+c) | From base‑height or Heron’s formula |
| Circle | All points equidistant from centre | (A = \pi r^2), (C = 2\pi r) | Integral of (\pi r^2) or limit of regular polygons |
| Regular Polygon | All sides & angles equal | (A = \frac{1}{4}n s^2 \cot!\frac{\pi}{n}) | Divide into (n) isosceles triangles |
| Ellipse | Two axes (a, b) | (A = \pi ab) | Integral of (\pi a b) by substitution |
2.1 Deriving the Circle Area (First Principles)
- Define area of a circle (A(r)) as a function of radius (r).
- Use the differential relation (\displaystyle \frac{dA}{dr} = 2\pi r) (circumference).
- Integrate: (\displaystyle A(r) = \int_0^r 2\pi t,dt = \pi r^2).
- Edge Case – For a degenerate circle ((r=0)), (A=0).
2.2 Edge Cases & Degeneracies
| Edge Case | Description | Result |
|---|---|---|
| Zero‑area polygons | All vertices collinear | (A=0) |
| Self‑intersecting polygons | “Star” shapes | Use signed area (Green’s theorem) |
| Tangent circles | Two circles touch externally | Intersection area = 0 |
| Coincident circles | Same centre, same radius | Intersection area = area of one circle |
3. Three‑Dimensional Figures
| Figure | Key Properties | General Formula | Derivation Sketch |
|---|---|---|---|
| Cube | 6 congruent squares | (V = a^3), (SA = 6a^2) | Volume = side × area of base |
| Cuboid | 6 rectangles | (V = lwh), (SA = 2(lw+lh+wh)) | Rectangular prisms |
| Cylinder | Circular base, height (h) | (V = \pi r^2 h), (SA = 2\pi r(h+r)) | Integrate disk area along height |
| Sphere | All points at distance (r) from centre | (V = \frac{4}{3}\pi r^3), (SA = 4\pi r^2) | Integrate disk area or use surface integral |
| Regular Tetrahedron | 4 equilateral triangles | (V = \frac{\sqrt{2}}{12}a^3), (SA = \sqrt{3}a^2) | Decompose into pyramids |
| Regular Octahedron | 8 equilateral triangles | (V = \frac{\sqrt{2}}{3}a^3), (SA = 2\sqrt{3}a^2) | Similar decomposition |
| **Regular Dodecahedron |