CDS 2027 - Paper 3: Elementary Mathematics cheat sheet for Mensuration! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Elementary Mathematics: Mensuration Cheat Sheet

Mensuration forms a crucial component of the CDS Elementary Mathematics syllabus, testing spatial understanding, geometric properties, and numerical computation across two-dimensional (2D) and three-dimensional (3D) figures.

1. Two-Dimensional Shapes (Area and Perimeter)

2D figures focus on region enclosed (Area) and boundary length (Perimeter).

• Triangles: For a triangle with sides \(a\), \(b\), \(c\) and semi-perimeter \(s = \frac{a+b+c}{2}\), the area is given by Heron's formula: \[Area = \sqrt{s(s-a)(s-b)(s-c)}\] For an equilateral triangle with side \(a\), \(Area = \frac{\sqrt{3}}{4}a^2\) and \(Height = \frac{\sqrt{3}}{2}a\).
• Quadrilaterals:
  • Rectangle: \(Area = l \times b\), \(Perimeter = 2(l+b)\), \(Diagonal = \sqrt{l^2+b^2}\)
  • Rhombus: \(Area = \frac{1}{2} d_1 d_2\), where \(d_1\) and \(d_2\) are diagonal lengths.
  • Trapezium: \(Area = \frac{1}{2} (a+b) h\), where \(a\) and \(b\) are parallel sides and \(h\) is height.
• Circles and Sectors: For radius \(r\), \(Area = \pi r^2\) and \(Circumference = 2\pi r\). For a sector with angle \(\theta\) (in degrees), \(Area = \frac{\theta}{360^{\circ}} \times \pi r^2\) and \(Arc\ Length = \frac{\theta}{360^{\circ}} \times 2\pi r\).

2. Three-Dimensional Solids (Surface Area and Volume)

3D solids evaluate total space occupied (Volume), lateral/curved surface area (LSA/CSA), and total surface area (TSA).

• Cylinder: \(Volume = \pi r^2 h\), \(CSA = 2\pi rh\), \(TSA = 2\pi r(h+r)\)
• Cone: Slant height \(l = \sqrt{r^2+h^2}\), \(Volume = \frac{1}{3}\pi r^2 h\), \(CSA = \pi rl\), \(TSA = \pi r(l+r)\)
• Sphere and Hemisphere: For a sphere, \(Volume = \frac{4}{3}\pi r^3\) and \(Surface\ Area = 4\pi r^2\). For a hemisphere, \(Volume = \frac{2}{3}\pi r^3\), \(CSA = 2\pi r^2\), and \(TSA = 3\pi r^2\).
• Frustum of a Cone: With radii \(R\), \(r\) and height \(h\), \(Volume = \frac{1}{3}\pi h(R^2 + r^2 + R r)\).