RRB PREP
NTPC ALP Group D

Geometry for RRB Exams

Lines & Angles

Basic Concepts

  • • Complementary Angles: Sum = 90°
  • • Supplementary Angles: Sum = 180°
  • • Linear Pair: Adjacent angles on a line sum to 180°
  • • Vertically Opposite Angles: Are equal

Parallel Lines

  • • Corresponding angles are equal
  • • Alternate interior angles are equal
  • • Sum of interior angles on same side = 180°

Triangles

Properties

  • • Sum of angles = 180°
  • • Exterior angle = Sum of two opposite interior angles
  • • Sum of any two sides > Third side
  • • Difference of any two sides < Third side

Centers of Triangle

  • • Centroid: Intersection of medians (divides median in 2:1)
  • • Incenter: Intersection of angle bisectors
  • • Circumcenter: Intersection of perpendicular bisectors
  • • Orthocenter: Intersection of altitudes

Circles

Theorems

  • • Perpendicular from center to chord bisects the chord
  • • Equal chords are equidistant from center
  • • Angle at center is double the angle at remaining part of circle
  • • Angles in same segment are equal
  • • Angle in semicircle is 90°
  • • Tangent is perpendicular to radius at point of contact

Mensuration (2D & 3D)

2D Formulas (Area)

  • • Triangle: ½ × base × height
  • • Equilateral Triangle: (√3/4)a²
  • • Rectangle: l × b
  • • Square: a²
  • • Circle: πr² (Circumference: 2πr)
  • • Parallelogram: base × height
  • • Rhombus: ½ × d₁ × d₂
  • • Trapezium: ½ × (a+b) × h

3D Formulas (Volume)

  • • Cube: a³ (TSA: 6a²)
  • • Cuboid: l×b×h (TSA: 2(lb+bh+hl))
  • • Cylinder: πr²h (TSA: 2πr(r+h))
  • • Cone: ⅓πr²h (TSA: πr(r+l))
  • • Sphere: (4/3)πr³ (TSA: 4πr²)
  • • Hemisphere: (2/3)πr³ (TSA: 3πr²)

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