13  Geometry and Mensuration

13.1 Introduction

Geometry studies shapes, sizes, and properties of figures.
Mensuration deals with measuring lengths, areas, and volumes.
Together they form one of the most important areas of quantitative aptitude.
Questions typically include properties of lines, angles, triangles, circles, quadrilaterals, polygons, and computation of areas, perimeters, surface areas, and volumes.

13.2 1) Lines and Angles

13.2.1 1.1 Basic Angle Facts

  • Straight line = 180°
  • Right angle = 90°
  • Complete angle = 360°

13.2.2 1.2 Angle Relationships

  • Vertically opposite angles = equal
  • Corresponding, alternate interior angles equal (when parallel lines cut by transversal)
  • Sum of angles in triangle = 180°

13.3 2) Triangles

13.3.1 2.1 Types

  • Equilateral: all sides equal, each angle = 60°
  • Isosceles: two sides equal
  • Right-angled: one angle = 90°

13.3.2 2.2 Properties

  • Exterior angle = sum of two opposite interior angles

  • Area (Heron’s formula):
    \[A = \sqrt{s(s-a)(s-b)(s-c)}\]
    where \(s=\tfrac{a+b+c}{2}\)

  • Pythagoras theorem: In right triangle, \(a^2+b^2=c^2\)

13.3.3 2.3 Special Results

  • In equilateral triangle of side \(a\):
    Height \(=\tfrac{\sqrt{3}}{2}a\),
    Area \(=\tfrac{\sqrt{3}}{4}a^2\)

13.4 3) Quadrilaterals and Polygons

13.4.1 3.1 Parallelogram

  • Opposite sides and angles equal.
  • Area = base × height.

13.4.2 3.2 Rectangle and Square

  • Rectangle: Area = \(l \times b\), Perimeter = \(2(l+b)\).
  • Square: Area = \(a^2\), Diagonal = \(a\sqrt{2}\).

13.4.3 3.3 Trapezium

  • Area = \(\tfrac{1}{2}(a+b)\cdot h\) where \(a,b\) are parallel sides.

13.4.4 3.4 General Polygon

  • Sum of interior angles = \((n-2)\times180°\).
  • Each interior angle (regular polygon) = \(\tfrac{(n-2)}{n}\times180°\).

13.5 4) Circles

13.5.1 4.1 Definitions

  • Radius, diameter, chord, tangent, secant, arc, sector, segment.

13.5.2 4.2 Properties

  • Angle at center = 2 × angle at circumference (same arc).
  • Angle in semicircle = 90°.
  • Tangent ⟂ radius at point of contact.

13.5.3 4.3 Lengths and Areas

  • Circumference = \(2\pi r\).
  • Area = \(\pi r^2\).
  • Arc length = \(\tfrac{\theta}{360} \cdot 2\pi r\).
  • Area of sector = \(\tfrac{\theta}{360} \cdot \pi r^2\).

13.6 5) Mensuration – 2D Figures

  • Rectangle: \(A=lb\), \(P=2(l+b)\)
  • Square: \(A=a^2\), \(P=4a\)
  • Triangle: \(A=\tfrac{1}{2}bh\)
  • Circle: \(A=\pi r^2\), \(C=2\pi r\)
  • Trapezium: \(A=\tfrac{1}{2}(a+b)h\)
  • Rhombus: \(A=\tfrac{1}{2}d_1d_2\)

13.7 6) Mensuration – 3D Solids

13.7.1 6.1 Cuboid

  • Volume = \(l\cdot b\cdot h\)
  • TSA = \(2(lb+bh+hl)\)
  • LSA = \(2h(l+b)\)

13.7.2 6.2 Cube

  • Volume = \(a^3\)
  • TSA = \(6a^2\)
  • Diagonal = \(a\sqrt{3}\)

13.7.3 6.3 Cylinder

  • Volume = \(\pi r^2h\)
  • CSA = \(2\pi rh\)
  • TSA = \(2\pi r(h+r)\)

13.7.4 6.4 Cone

  • Volume = \(\tfrac{1}{3}\pi r^2h\)
  • Slant height \(l=\sqrt{r^2+h^2}\)
  • CSA = \(\pi rl\)

13.7.5 6.5 Sphere & Hemisphere

  • Sphere: \(V=\tfrac{4}{3}\pi r^3\), \(SA=4\pi r^2\)
  • Hemisphere: \(V=\tfrac{2}{3}\pi r^3\), \(CSA=2\pi r^2\), \(TSA=3\pi r^2\)

13.8 7) Solved Examples

Ex 1 Find area of equilateral triangle side 12.
\(A=\tfrac{\sqrt{3}}{4}\cdot12^2=36\sqrt{3}\).

Ex 2 Diagonal of rectangle = 25, breadth=15. Find length.
\(l=\sqrt{25^2-15^2}=20\).

Ex 3 A circle radius 7 cm. Find circumference.
\(C=2\pi r=44\) cm approx.

Ex 4 Volume of cylinder radius 7 cm, height 10 cm.
\(V=\pi r^2h=1540\) cm³.

Ex 5 Find CSA of cone radius 3 cm, height 4 cm.
\(l=\sqrt{3^2+4^2}=5\), CSA=\(\pi rl=15\pi\) cm².

13.9 8) Practice Set

13.9.1 Set A – Geometry

  1. Angles of triangle are in ratio 2:3:4. Find angles.
  2. Find area of square whose diagonal=10.
  3. In a trapezium parallel sides 8 cm and 12 cm, height=6 cm. Find area.
  4. Number of diagonals in 12-sided polygon.
  5. In a circle radius 14, find length of arc subtending 90°.

13.9.2 Set B – Mensuration

  1. Volume of cube edge 8 cm.
  2. Find TSA of cuboid 5×4×3.
  3. A sphere radius 7 cm is melted into cones radius 7, height 3. Number of cones?
  4. CSA of cylinder=440, height=10. Find radius.
  5. A cone has height=24, radius=7. Find slant height and CSA.

13.10 9) Common Mistakes

  • Mixing perimeter with area.
  • Using diameter instead of radius in formulas.
  • Forgetting units (cm², cm³).
  • Confusing CSA, TSA, and volume.
  • Not recognizing that trapezium/triangle formulas reduce to simpler cases.

13.11 Summary

  • Geometry: properties of lines, angles, polygons, circles.
  • Mensuration: formulas for 2D areas, 3D volumes, and surface areas.
  • Core exam skill: memorize standard formulas and practice speed calculation.
  • Visualization and diagram drawing are key for solving geometry word problems.