Mensuration Formula Sheet - IBPS PO Prelims

1. Exam Context & PYQ Weightage Analysis

In the IBPS PO Prelims Quantitative Aptitude section, Mensuration typically accounts for 1 to 3 direct questions. Furthermore, concepts of 2D and 3D mensuration frequently appear in Data Interpretation (DI) sets, particularly Caselet DI and Mensuration-based Radar/Bar charts.

NTA / IBPS Exam Pattern Trends: Questions usually involve a combination of two shapes (e.g., a sphere melted into a cylinder, or a circle inscribed in a square) or ratio-based parameter variations. Memorizing direct geometric relations and efficiency in calculation are essential for attempting these questions within the tight sectional timing.

2. Chapter 1: 2D Shapes (Plane Mensuration)

2D Mensuration deals with boundary lengths (perimeter/circumference) and surface coverage (area) of flat two-dimensional figures.

2.1 Triangles

For any triangle with base \(b\) and height \(h\):

\[A = \frac{1}{2} \times b \times h\]

When all three sides \(a\), \(b\), and \(c\) are known, use Heron's Formula:

\[A = \sqrt{s(s-a)(s-b)(s-c)}\]

where the semi-perimeter \(s = \frac{a+b+c}{2}\).

For an Equilateral Triangle with side length \(a\):

• Area: \[A = \frac{\sqrt{3}}{4} a^2\]
• Height: \[h = \frac{\sqrt{3}}{2} a\]

2.2 Quadrilaterals

• Rectangle: Area \(= l \times b\), Perimeter \(= 2(l + b)\), Diagonal \(= \sqrt{l^2 + b^2}\)
• Square: Area \(= a^2\), Perimeter \(= 4a\), Diagonal \(= a\sqrt{2}\)
• Parallelogram: Area \(= b \times h\)
• Rhombus: Area \(= \frac{1}{2} \times d_1 \times d_2\) (where \(d_1\) and \(d_2\) are diagonal lengths)
• Trapezium: Area \(= \frac{1}{2}(a + b) \times h\) (where \(a\) and \(b\) are parallel sides)

2.3 Circles, Arcs & Sectors

For a circle of radius \(r\):

• Area: \[A = \pi r^2\]
• Circumference: \[C = 2\pi r\]
• Arc Length (\(l\)) subtending angle \(\theta\) at center: \[l = \frac{\theta}{360^\circ} \times 2\pi r\]
• Area of Sector: \[A = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{1}{2} \cdot l \cdot r\]

3. Chapter 2: 3D Shapes (Solid Mensuration)

3D Mensuration covers volumetric capacity, Curved/Lateral Surface Area (CSA/LSA), and Total Surface Area (TSA) of three-dimensional solids.

3.1 Cubes & Cuboids

Cube (Edge length \(a\)):

• Total Surface Area (TSA): \[6a^2\]
• Curved / Lateral Surface Area (LSA): \[4a^2\]
• Volume: \[a^3\]
• Space Diagonal: \[a\sqrt{3}\]

Cuboid (Length \(l\), Width \(b\), Height \(h\)):

• TSA: \[2(lb + bh + hl)\]
• LSA (Area of 4 walls): \[2(l + b)h\]
• Volume: \[l \times b \times h\]
• Space Diagonal: \[\sqrt{l^2 + b^2 + h^2}\]

3.2 Cylinders & Cones

Right Circular Cylinder (Radius \(r\), Height \(h\)):

• Curved Surface Area (CSA): \[2\pi rh\]
• Total Surface Area (TSA): \[2\pi r(r + h)\]
• Volume: \[\pi r^2 h\]

Right Circular Cone (Radius \(r\), Height \(h\)):

• Slant Height (\(l\)): \[l = \sqrt{r^2 + h^2}\]
• CSA: \[\pi rl\]
• TSA: \[\pi r(r + l)\]
• Volume: \[\frac{1}{3}\pi r^2 h\]

3.3 Spheres & Hemispheres

Sphere (Radius \(r\)):

• Surface Area: \[4\pi r^2\]
• Volume: \[\frac{4}{3}\pi r^3\]

Hemisphere (Radius \(r\)):

• Curved Surface Area (CSA): \[2\pi r^2\]
• Total Surface Area (TSA): \[3\pi r^2\]
• Volume: \[\frac{2}{3}\pi r^3\]

4. Frequently Asked Questions (People Also Ask)