Mensuration Formula Cheat Sheet - CDS 2027 - Paper 3: Elementary Mathematics

1. Exam Analysis & Weightage (CDS NTA/UPSC Pattern)

In the CDS Elementary Mathematics Paper, Mensuration typically carries a weightage of 12% to 15% of the total questions (roughly 12-15 questions out of 100). The questions range from direct formula application to multi-stage conceptual problems combining 2D projections with 3D volumes (e.g., converting a sphere into a cylinder or frustum calculations).

PYQ Analysis: Past Year Questions highlight a heavy emphasis on equilateral triangle properties, rhombus diagonal relationships, frustum surface areas, and ratio-based volume transformations.

2. Chapter 1: 2D Geometry Formulas

2D Mensuration focuses on the calculation of perimeter, semi-perimeter, area, and diagonal lengths for two-dimensional closed figures.

Triangles

• General Triangle Area: \(\text{Area} = \frac{1}{2} \cdot b \cdot h\)
• Heron's Formula: \(A = \sqrt{s(s-a)(s-b)(s-c)}\) where semi-perimeter \(s = \frac{a+b+c}{2}\)
• Equilateral Triangle: Area \(A = \frac{\sqrt{3}}{4} a^2\), Height \(h = \frac{\sqrt{3}}{2} a\)

Quadrilaterals

• Rectangle: Area \(A = l \cdot w\), Perimeter \(P = 2(l + w)\), Diagonal \(d = \sqrt{l^2 + w^2}\)
• Square: Area \(A = a^2\), Perimeter \(P = 4a\), Diagonal \(d = a\sqrt{2}\)
• Parallelogram: Area \(A = b \cdot h\)
• Rhombus: Area \(A = \frac{1}{2} d_1 d_2\), Perimeter \(P = 4a\), Side \(a = \frac{1}{2}\sqrt{d_1^2 + d_2^2}\)
• Trapezoid: Area \(A = \frac{1}{2} (a + b) \cdot h\)

Circles & Circular Sectors

• Circle: Area \(A = \pi r^2\), Circumference \(C = 2\pi r = \pi d\)
• Circular Sector Arc Length: \(L = \frac{\theta}{360^\circ} \cdot 2\pi r\)
• Circular Sector Area: \(A = \frac{\theta}{360^\circ} \cdot \pi r^2 = \frac{1}{2} L r\)

3. Chapter 2: 3D Solid Geometry Formulas

3D Mensuration encompasses Lateral/Curved Surface Area (LSA/CSA), Total Surface Area (TSA), and Volume (V) of three-dimensional solids.

Key Formulas Overview

• Cube: \(\text{LSA} = 4a^2\), \(\text{TSA} = 6a^2\), \(V = a^3\)
• Cuboid: \(\text{LSA} = 2h(l+w)\), \(\text{TSA} = 2(lw+wh+hl)\), \(V = l \cdot w \cdot h\)
• Cylinder: \(\text{CSA} = 2\pi r h\), \(\text{TSA} = 2\pi r (r + h)\), \(V = \pi r^2 h\)
• Cone: \(\text{CSA} = \pi r l\) (where slant height \(l = \sqrt{r^2+h^2}\)), \(\text{TSA} = \pi r(r + l)\), \(V = \frac{1}{3} \pi r^2 h\)
• Sphere: \(\text{CSA} = 4\pi r^2\), \(\text{TSA} = 4\pi r^2\), \(V = \frac{4}{3} \pi r^3\)
• Hemisphere: \(\text{CSA} = 2\pi r^2\), \(\text{TSA} = 3\pi r^2\), \(V = \frac{2}{3} \pi r^3\)
• Frustum of a Cone: \(\text{CSA} = \pi l (R + r)\), \(\text{TSA} = \text{CSA} + \pi R^2 + \pi r^2\), \(V = \frac{1}{3}\pi h (R^2 + r^2 + R r)\)

4. Chapter 3: Formula Reference Table

5. Frequently Asked Questions (People Also Ask)

What is the weightage of Mensuration in CDS Paper 3 Elementary Mathematics?

Mensuration carries approximately 12 to 15 marks out of 100 in the CDS Paper 3 examination, making it one of the highest-yielding topics alongside Trigonometry and Geometry.

What is the difference between Lateral Surface Area and Total Surface Area?

Lateral Surface Area (LSA) or Curved Surface Area (CSA) measures the area of the sides excluding the top and bottom bases. Total Surface Area (TSA) includes the LSA plus the areas of both top and bottom bases.

How do you calculate the slant height of a Frustum?

The slant height \(l\) of a frustum with height \(h\) and base radii \(R\) and \(r\) is given by the formula \(l = \sqrt{h^2 + (R - r)^2}\).