Geometry and Mensuration

Master geometry and mensuration concepts for RRB exam preparation with comprehensive coverage of formulas, properties, and problem-solving techniques.

Basic Geometry

Points, Lines and Angles

Fundamental Concepts

• Point: Location with no size or dimensions
• Line: Collection of points extending infinitely in both directions
• Line Segment: Part of a line with two endpoints
• Ray: Part of a line with one endpoint extending infinitely

Types of Lines

• Parallel Lines: Lines that never intersect
• Perpendicular Lines: Lines that intersect at 90°
• Intersecting Lines: Lines that cross each other
• Skew Lines: Lines that don’t intersect and aren’t parallel (3D)

Angles

Classification by Measure

• Acute Angle: < 90°
• Right Angle: = 90°
• Obtuse Angle: > 90° but < 180°
• Straight Angle: = 180°
• Reflex Angle: > 180° but < 360°
• Complete Angle: = 360°

Angle Relationships

• Complementary Angles: Sum = 90°
• Supplementary Angles: Sum = 180°
• Adjacent Angles: Share a common arm
• Vertical Angles: Opposite angles formed by intersecting lines

Triangles

Triangle Classification

By Sides

• Equilateral Triangle: All sides equal, all angles 60°
• Isosceles Triangle: Two sides equal, two base angles equal
• Scalene Triangle: All sides different, all angles different

By Angles

• Acute Triangle: All angles < 90°
• Right Triangle: One angle = 90°
• Obtuse Triangle: One angle > 90°

Triangle Properties

Sum of Angles

• Sum of all angles = 180°

Pythagorean Theorem

• For right triangles: a² + b² = c²
• Where ‘c’ is the hypotenuse (longest side)

Area and Perimeter

• Area = (1/2) × base × height
• Perimeter = sum of all sides

Special Right Triangles

• 30-60-90 Triangle: Sides in ratio 1:√3:2
• 45-45-90 Triangle: Sides in ratio 1:1:√2

Quadrilaterals

Types of Quadrilaterals

Parallelogram

• Properties: Opposite sides parallel and equal
• Area = base × height
• Perimeter = 2 × (length + width)

Rectangle

• Properties: Parallelogram with all angles = 90°
• Area = length × width
• Perimeter = 2 × (length + width)
• Diagonal = √(length² + width²)

Square

• Properties: Rectangle with all sides equal
• Area = side²
• Perimeter = 4 × side
• Diagonal = side × √2

Rhombus

• Properties: Parallelogram with all sides equal
• Area = (1/2) × d₁ × d₂ (product of diagonals)
• Perimeter = 4 × side

Trapezium (Trapezoid)

• Properties: One pair of parallel sides
• Area = (1/2) × (sum of parallel sides) × height
• Perimeter = sum of all sides

Circles

Circle Properties

Basic Definitions

• Circle: Set of points equidistant from center
• Radius: Distance from center to any point on circle
• Diameter: Longest chord through center (2 × radius)
• Circumference: Perimeter of circle
• Chord: Line segment connecting two points on circle
• Arc: Part of circumference
• Sector: Region bounded by two radii and an arc

Important Formulas

Circumference

• C = 2πr or πd
• Where π ≈ 3.14 or 22/7

Area

• A = πr²

Area of Sector

• Area = (θ/360°) × πr²
• Where θ is the central angle

Length of Arc

• Length = (θ/360°) × 2πr

Circle Theorems

Important Theorems

• Equal chords subtend equal angles at center
• Angle in a semicircle = 90°
• Angles in same segment are equal
• Perpendicular from center to chord bisects chord

3D Geometry

Cube

• Surface Area = 6a²
• Volume = a³
• Diagonal = a√3

Cuboid

• Surface Area = 2(lw + lh + wh)
• Volume = l × w × h
• Diagonal = √(l² + w² + h²)

Cylinder

• Curved Surface Area = 2πrh
• Total Surface Area = 2πr(r + h)
• Volume = πr²h

Cone

• Slant Height = √(r² + h²)
• Curved Surface Area = πrl
• Total Surface Area = πr(l + r)
• Volume = (1/3)πr²h

Sphere

• Surface Area = 4πr²
• Volume = (4/3)πr³

Hemisphere

• Curved Surface Area = 2πr²
• Total Surface Area = 3πr²
• Volume = (2/3)πr³

Mensuration

Area Formulas

2D Shapes

Square

• Area = side²

Rectangle

• Area = length × width

Triangle

• Area = (1/2) × base × height

Circle

• Area = πr²

Parallelogram

• Area = base × height

Trapezium

• Area = (1/2) × (sum of parallel sides) × height

Rhombus

• Area = (1/2) × product of diagonals

Perimeter Formulas

2D Shapes

Square

• Perimeter = 4 × side

Rectangle

• Perimeter = 2 × (length + width)

Triangle

• Perimeter = sum of all sides

Circle

• Perimeter = 2πr

Volume Formulas

3D Shapes

Cube

• Volume = side³

Cuboid

• Volume = length × width × height

Cylinder

• Volume = πr²h

Cone

• Volume = (1/3)πr²h

Sphere

• Volume = (4/3)πr³

Hemisphere

• Volume = (2/3)πr³

Surface Area Formulas

3D Shapes

Cube

• Surface Area = 6 × side²

Cuboid

• Surface Area = 2(lw + lh + wh)

Cylinder

• Curved Surface Area = 2πrh
• Total Surface Area = 2πr(r + h)

Cone

• Curved Surface Area = πrl
• Total Surface Area = πr(l + r)

Sphere

• Surface Area = 4πr²

Hemisphere

• Curved Surface Area = 2πr²
• Total Surface Area = 3πr²

Similarity and Congruency

Similar Figures

Properties of Similar Figures

• Corresponding angles are equal
• Corresponding sides are proportional
• Areas are in ratio of squares of corresponding sides
• Volumes are in ratio of cubes of corresponding sides

Similarity Ratios

• Linear Ratio: k (scale factor)
• Area Ratio: k²
• Volume Ratio: k³

Congruent Figures

Triangle Congruency Rules

• SSS: Side-Side-Side
• SAS: Side-Angle-Side
• ASA: Angle-Side-Angle
• RHS: Right angle-Hypotenuse-Side (for right triangles)

Coordinate Geometry

Distance Formula

• Distance between points (x₁, y₁) and (x₂, y₂): d = √[(x₂-x₁)² + (y₂-y₁)²]

Section Formula

• Point dividing line joining (x₁, y₁) and (x₂, y₂) in ratio m:n: x = (mx₂ + nx₁)/(m+n) y = (my₂ + ny₁)/(m+n)

Area of Triangle

• Triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃): Area = (1/2) |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

Quick Reference

Important Values

• π ≈ 3.14 or 22/7
• √2 ≈ 1.414
• √3 ≈ 1.732
• √5 ≈ 2.236

Common Conversions

• 1 cm = 10 mm
• 1 m = 100 cm = 1000 mm
• 1 km = 1000 m
• 1 cm² = 100 mm²
• 1 m² = 10,000 cm² = 1,000,000 mm²
• 1 km² = 1,000,000 m²

Problem-Solving Tips

Strategy for Geometry Problems

1. Draw Diagram: Always draw a clear diagram
2. Label Everything: Mark all given information
3. Identify What’s Given: Understand the given information
4. Look for Relationships: Identify similar or congruent figures
5. Apply Formulas: Use appropriate formulas
6. Check Units: Ensure consistent units

Common Mistakes to Avoid

1. Incorrect Formula Selection: Use correct formula for the shape
2. Unit Mistakes: Keep units consistent
3. Calculation Errors: Double-check calculations
4. Diagram Errors: Draw accurate diagrams
5. Formula Substitution: Substitute values correctly

Practice Questions

Question 1

Find the area of a circle with radius 7 cm.

Question 2

A rectangular field has dimensions 20m by 15m. Find its perimeter.

Question 3

Find the volume of a cylinder with radius 3cm and height 10cm.

Question 4

The sides of a triangle are 5cm, 12cm, and 13cm. Find its area.

Question 5

Find the surface area of a cube with side 4cm.

Question 6

A cone has radius 5cm and slant height 13cm. Find its volume.

Question 7

Find the area of a trapezium with parallel sides 8cm and 12cm, and height 5cm.

Question 8

The diameter of a sphere is 14cm. Find its surface area.

Question 9

A cuboid has dimensions 10cm × 8cm × 6cm. Find its volume.

Question 10

Find the area of a sector with radius 6cm and central angle 60°.

Mental Math Tips

Quick Calculations

• Square of numbers ending with 5: Use pattern (n5)² = n(n+1)25
• Multiplication by 11: For 2-digit numbers, add digits and place in middle
• Area of right triangle: Half of rectangle area
• Circumference estimation: 3 × diameter for quick estimate

Approximation Methods

• π ≈ 3: For rough calculations
• √2 ≈ 1.4: For quick estimates
• √3 ≈ 1.7: For quick estimates
• Volume estimation: Use simpler shapes for approximation

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