Mensuration – Advanced University‑Level Notes
(MIT‑style rigorous treatment for the Duolingo English Test – Quantitative Aptitude)
Purpose: Equip you with a deep, principled understanding of area, volume, surface area, and related concepts so you can solve any Mensuration problem, including edge cases and trick questions that appear in high‑level aptitude tests.
Prerequisites: Basic geometry, algebra, trigonometry, limits, and a touch of calculus.
1. Foundations
| Concept | Symbol | Definition | Units | Typical Use |
|---|---|---|---|---|
| Area | (A) | Measure of the size of a two‑dimensional figure. | (\text{cm}^2,\ \text{m}^2,\ldots) | Surface area of plane figures, cross‑sectional area |
| Volume | (V) | Measure of the amount of space occupied by a three‑dimensional figure. | (\text{cm}^3,\ \text{m}^3,\ldots) | Capacity, mass (if density known) |
| Surface Area | (S) | Total area of all faces of a 3‑D solid. | (\text{cm}^2,\ \text{m}^2,\ldots) | Paint needed, heat transfer |
| Perimeter | (P) | Length of the boundary of a plane figure. | (\text{cm},\ \text{m},\ldots) | Edge length of polygons |
First Principle Insight:
Area is the limit of the sum of infinitesimal rectangles that cover the shape.
Volume is the limit of the sum of infinitesimal boxes that fill the solid.
2. Plane Figures
2.1 Polygons
| Polygon | Formula | Derivation Notes |
|---|---|---|
| Triangle | (A = \frac12 , b , h) | From base × height divided by 2 (area of parallelogram). |
| (A = \sqrt{s(s-a)(s-b)(s-c)}) (Heron) | Derived by considering two right triangles inside, using Pythagoras and semiperimeter (s = \frac{a+b+c}{2}). | |
| (A = \frac{abc}{4R}) (circumradius (R)) | From (a = 2R\sin A) etc. | |
| Rectangle | (A = l \cdot w) | Directly from grid interpretation. |
| Parallelogram | (A = b \cdot h) | From parallelogram as two congruent triangles. |
| Rhombus | (A = \frac12 d_1 d_2) | Diagonals bisect at right angles; area = area of four right triangles. |
| Regular Polygon | (A = \frac{1}{4} n s^2 \cot(\pi/n)) | Derived by dividing into (n) isosceles triangles and summing. |
| Circle | (A = \pi r^2) | Integral: (\displaystyle \int_0^r 2\pi x,dx). |
Edge Cases
| Shape | Degenerate Condition | Area |
|---|---|---|
| Triangle | (a+b=c) (collinear points) | (0) |
| Rectangle | (l=0) or (w=0) | (0) |
| Circle | (r=0) | (0) |
2.2 Conic Sections
| Conic | General Equation | Area |
|---|---|---|
| Ellipse | (\frac{x^2}{a^2} + \frac{y^2}{b^2}=1) | (A = \pi a b) |
| Parabola (bounded by a chord) | ((x-h)^2 = 4p(y-k)) | Use shoelace or integral: (A = \frac{1}{2} \times \text{base} \times \text{height}). |
| Hyperbola | (\frac{x^2}{a^2} - \frac{y^2}{b^2}=1) | Area between asymptotes is infinite; finite area only when bounded by lines. |
3. Solid Figures
3.1 Prisms & Pyramids
| Solid | Volume Formula | Surface Area Formula |
|---|---|---|
| Right Prism | (V = A_{\text{base}} \cdot h) | (S = 2 A_{\text{base}} + P_{\text{base}} \cdot h) |
| Right Pyramid | (V = \frac13 A_{\text{base}} \cdot h) | (S = A_{\text{base}} + \sum \text{lateral triangular areas}) |
| Frustum | (V = \frac{h}{3}(A_1 + A_2 + \sqrt{A_1 A_2})) | (S = A_1 + A_2 + \pi (r_1 + r_2) \ell) (for conical frustum) |
Derivation: Integrate the cross‑sectional area along the axis of the solid. For prisms, (A_{\text{base}}) is constant. For pyramids, area scales quadratically with distance from apex: (A(x) = A_{\text{base}}\left(\frac{x}{h}\right)^2).
3.2 Cylinders & Cones
| Solid | Volume | Surface Area |
|---|---|---|
| Right Cylinder | (V = \pi r^2 h) | (S = 2\pi r h + 2\pi r^2) |
| Right Cone | (V = \frac13 \pi r^2 h) | (S = \pi r (r + \sqrt{r^2 + h^2})) |
| Sphere | (V = \frac43 \pi r^3) | (S = 4\pi r^2) |
**Der