Mensuration 2D and 3D Cheat Sheet - CDS 2027 Notes & Formula Reference Guide
Table of Contents
1. CDS 2027 Mensuration: Exam Context & Weightage Analysis
Exam Context: Mensuration is one of the highest-weightage topics in the Elementary Mathematics paper for the Combined Defence Services (CDS) examination conducted by UPSC. Historically, candidates can expect 12 to 18 questions purely from 2D and 3D Mensuration in CDS 2027.
UPSC Exam Pattern & PYQ Insights: UPSC frequently combines Mensuration concepts with Geometry, Trigonometry, and Ratio-Proportion. Direct conceptual formula application is common, alongside questions testing structural modifications (e.g., melting solids into new shapes, slicing cones into frustums, and percentage changes in dimensions).
2. Chapter 1: Two-Dimensional Shapes (2D Geometry)
1.1 Triangles
Triangles form the baseline for planar geometry and multi-sided polygons.
| Triangle Type | Property / Parameter | Formula |
|---|---|---|
| General Triangle | Area (Base & Height) | \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) |
| General Triangle | Heron's Formula | \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \frac{a+b+c}{2}\) |
| Equilateral Triangle | Area | \(\text{Area} = \frac{\sqrt{3}}{4}a^2\) |
1.2 Rectangles & Squares
| Shape | Area | Perimeter | Diagonal |
|---|---|---|---|
| Rectangle | \(l \times b\) | \(2(l + b)\) | \(\sqrt{l^2 + b^2}\) |
| Square | \(a^2\) | \(4a\) | \(a\sqrt{2}\) |
1.3 Parallelogram, Rhombus & Trapezium
| Shape | Area Formula | Perimeter / Property |
|---|---|---|
| Parallelogram | \(\text{base} \times \text{height}\) | \(2(\text{side}_1 + \text{side}_2)\) |
| Rhombus | \(\frac{1}{2} \times d_1 \times d_2\) | \(4a\) |
| Trapezium | \(\frac{1}{2} \times (a + b) \times h\) | Sum of all 4 outer sides |
1.4 Circles & Sectors
- Circle Area: \(\text{Area} = \pi r^2\)
- Circumference: \(\text{Perimeter} = 2\pi r\)
- Sector Area: \(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\)
- Arc Length: \(\text{Length} = \frac{\theta}{360^\circ} \times 2\pi r\)
3. Chapter 2: Three-Dimensional Shapes (3D Geometry)
2.1 Cuboid & Cube
| Solid | Lateral / Curved Surface Area | Total Surface Area (TSA) | Volume | Diagonal |
|---|---|---|---|---|
| Cuboid | \(2(l + b)h\) | \(2(lb + bh + hl)\) | \(l \times b \times h\) | \(\sqrt{l^2 + b^2 + h^2}\) |
| Cube | \(4a^2\) | \(6a^2\) | \(a^3\) | \(a\sqrt{3}\) |
2.2 Right Circular Cylinder & Right Circular Cone
| Solid | Slant Height (\(l\)) | CSA | TSA | Volume |
|---|---|---|---|---|
| Cylinder | N/A | \(2\pi rh\) | \(2\pi r(r + h)\) | \(\pi r^2 h\) |
| Cone | \(l = \sqrt{r^2 + h^2}\) | \(\pi rl\) | \(\pi r(r + l)\) | \(\frac{1}{3}\pi r^2 h\) |
2.3 Sphere & Hemisphere
| Solid | Curved Surface Area (CSA) | Total Surface Area (TSA) | Volume |
|---|---|---|---|
| Sphere | \(4\pi r^2\) | \(4\pi r^2\) | \(\frac{4}{3}\pi r^3\) |
| Hemisphere | \(2\pi r^2\) | \(3\pi r^2\) | \(\frac{2}{3}\pi r^3\) |
4. Chapter 3: Advanced Polygons, Frustum & Units Conversion
3.1 Polygon Formulas
- Sum of interior angles: \((n-2) \times 180^\circ\)
- Each interior angle (regular polygon): \(\frac{(n-2)180^\circ}{n}\)
- Number of diagonals: \(\frac{n(n-3)}{2}\)
3.2 Frustum of a Cone
When a right circular cone is cut parallel to its base, the lower portion forms a frustum of height \(h\), top radius \(r\), and bottom radius \(R\):
- Slant height (\(l\)): \(l = \sqrt{h^2 + (R-r)^2}\)
- Curved Surface Area (CSA): \(\text{CSA} = \pi(R+r)l\)
- Total Surface Area (TSA): \(\text{TSA} = \pi(R+r)l + \pi R^2 + \pi r^2\)
- Volume: \(\text{Volume} = \frac{1}{3}\pi h(R^2 + r^2 + Rr)\)
3.3 Standard Unit Conversions
- \(1 \text{ Litre} = 1000 \text{ cm}^3\)
- \(1 \text{ m}^3 = 10^6 \text{ cm}^3 = 1000 \text{ Litres}\)
- \(1 \text{ hectare} = 10,000 \text{ m}^2\)
5. People Also Ask (FAQ)
Q1: How many questions are asked from Mensuration in the CDS exam?
Typically, 12 to 18 questions are asked from 2D and 3D Mensuration combined in the CDS Elementary Mathematics paper. Mastering this topic guarantees significant scoring potential.
Q2: What is the main difference between CSA and TSA for a Hemisphere?
Curved Surface Area (CSA) accounts only for the curved dome portion (\(2\pi r^2\)), whereas Total Surface Area (TSA) includes the flat circular top surface, giving \(2\pi r^2 + \pi r^2 = 3\pi r^2\).
Q3: Where can I practice previous year questions (PYQs) for CDS Mensuration?
You can practice over 1 Lakh+ CDS, NDA, and AFCAT Previous Year Questions with full solutions on ExamBhai.com and access comprehensive notes on Telegram.
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