Quadrilaterals — Class 8 Mathematics (Ganita Prakash)
"Take any closed shape with four straight sides — its four angles always sum to 360°. From this single rule, the entire world of quadrilaterals unfolds."
1. About the Chapter
'Quadrilaterals' studies four-sided polygons — their types, properties, and construction. After learning about triangles in Class 7, you now extend to richer shapes that fill the real world: walls, screens, kites, diamonds, parallelogram-shaped fields.
Key Idea
Every quadrilateral has 4 sides, 4 angles, and 2 diagonals. Its angle sum is always 360°.
The classification tree:
- Quadrilateral
- Trapezium (one pair of parallel sides)
- Parallelogram (two pairs of parallel sides)
- Rectangle (parallelogram + all 90° angles)
- Rhombus (parallelogram + all sides equal)
- Square (rectangle + rhombus)
- Kite (two pairs of adjacent sides equal)
2. Basic Properties of a Quadrilateral
Definition
A quadrilateral is a closed figure bounded by 4 straight line segments.
Components
- 4 sides (edges)
- 4 vertices (corners)
- 4 interior angles
- 2 diagonals (lines connecting opposite vertices)
The 360° Property (CRITICAL)
Sum of all interior angles of a quadrilateral = 360°.
Proof outline: Any quadrilateral can be divided into 2 triangles by a diagonal. Each triangle has angle sum 180°. So quadrilateral = 180° × 2 = 360°. ✓
Convex vs Concave
- Convex quadrilateral: all interior angles < 180°
- Concave quadrilateral: one interior angle > 180° (like a dart or arrowhead shape)
In Class 8 we mostly study convex quadrilaterals.
3. Types of Quadrilaterals — A Systematic Study
3.1 Trapezium (US: Trapezoid)
Definition: A quadrilateral with EXACTLY ONE pair of parallel sides.
Properties:
- Only one pair of parallel sides (called bases)
- The other two sides are NON-parallel
- Isosceles trapezium: non-parallel sides are equal
- Diagonals are equal in isosceles trapezium
Example: A simple ramp shape; a school bench profile.
3.2 Parallelogram
Definition: A quadrilateral with BOTH pairs of opposite sides parallel.
Properties (memorise all):
- Opposite sides are equal (AB = CD; AD = BC)
- Opposite sides are parallel (AB || CD; AD || BC)
- Opposite angles are equal (∠A = ∠C; ∠B = ∠D)
- Consecutive angles are supplementary (∠A + ∠B = 180°)
- Diagonals bisect each other (the point where they cross divides each diagonal exactly in half)
- Each diagonal divides the parallelogram into two congruent triangles
Example: A leaning bookshelf.
3.3 Rectangle
Definition: A parallelogram with ALL FOUR ANGLES equal to 90°.
Properties:
- All properties of a parallelogram, PLUS:
- All angles = 90°
- Diagonals are EQUAL (AC = BD)
- Diagonals bisect each other (from parallelogram property)
Example: A door, a TV screen, a book.
3.4 Rhombus
Definition: A parallelogram with ALL FOUR SIDES equal.
Properties:
- All properties of a parallelogram, PLUS:
- All sides equal (AB = BC = CD = DA)
- Diagonals are PERPENDICULAR to each other (cross at 90°)
- Diagonals bisect each other (from parallelogram)
- Diagonals bisect the angles at vertices
Example: A traditional diamond shape on a playing card.
3.5 Square
Definition: A rectangle with all sides equal. (Equivalently: a rhombus with all angles 90°.)
Properties (all of rectangle AND rhombus):
- All sides equal
- All angles = 90°
- Diagonals equal in length
- Diagonals perpendicular
- Diagonals bisect each other
- Diagonals bisect vertex angles (so each angle becomes 45°)
Example: A chessboard square; a Rubik's cube face.
3.6 Kite
Definition: A quadrilateral with TWO PAIRS OF ADJACENT (consecutive) SIDES EQUAL.
Properties:
- Two pairs of consecutive equal sides
- One pair of opposite angles is equal (where the unequal sides meet)
- Diagonals are perpendicular
- One diagonal bisects the other
Example: An actual flying kite; certain gemstones.
4. The Quadrilateral Family Tree
Quadrilateral
|
___________________
| | |
Trapezium Kite Parallelogram
|
____________
| |
Rectangle Rhombus
| |
___________
|
Square
A square is at the bottom — it inherits properties from BOTH rectangle and rhombus.
5. Angle Sum and Exterior Angles
Interior Angle Sum
Sum = 360° for any quadrilateral.
If three angles are given, you can find the fourth: ∠D = 360° − (∠A + ∠B + ∠C).
Exterior Angle Sum
Sum of exterior angles (one at each vertex) of a quadrilateral = 360°.
In fact, sum of exterior angles of ANY polygon = 360°.
6. Diagonals — The Inner Lines
A quadrilateral has 2 diagonals (connecting opposite vertices).
Key facts:
- Parallelogram: diagonals BISECT each other
- Rectangle: diagonals are EQUAL in length
- Rhombus: diagonals are PERPENDICULAR
- Square: diagonals are EQUAL + PERPENDICULAR + bisect angles
- Kite: diagonals are PERPENDICULAR (one bisects the other)
- Isosceles trapezium: diagonals are EQUAL
7. Construction of Quadrilaterals (5 Cases)
To uniquely construct a quadrilateral, you need 5 measurements in suitable combinations. Here are the standard cases:
Case 1: 4 sides + 1 diagonal
Given AB, BC, CD, DA, and AC.
Procedure:
- Draw AC (length given).
- With A as centre, radius AB, draw an arc.
- With C as centre, radius BC, draw an arc; intersection = B.
- With A as centre, radius AD, draw an arc on opposite side of AC.
- With C as centre, radius CD, draw an arc; intersection = D.
- Join AB, BC, CD, DA.
Case 2: 3 sides + 2 diagonals
Given AB, BC, CD, and AC, BD.
Case 3: 2 adjacent sides + 3 angles
Given AB, BC, and ∠A, ∠B, ∠C.
Case 4: 3 sides + 2 included angles
Given AB, BC, CD, and ∠B, ∠C.
Case 5: Special types — rectangle, square, rhombus (need fewer measurements)
For a square: just one side is enough! For a rectangle: two adjacent sides. For a rhombus: one side + one diagonal (or side + one angle). For a parallelogram: two adjacent sides + one included angle.
8. Worked Examples
Example 1: Find Missing Angle
A quadrilateral has angles 70°, 110°, 80°. Find the fourth angle.
- Sum = 360°.
- 4th angle = 360° − (70° + 110° + 80°) = 360° − 260° = 100° ✓
Example 2: Parallelogram Properties
In parallelogram ABCD, ∠A = 65°. Find ∠B, ∠C, ∠D.
- ∠C = 65° (opposite angles equal)
- ∠B = ∠D = 180° − 65° = 115° (consecutive angles supplementary)
Example 3: Rectangle Diagonal
In rectangle ABCD with AB = 6 cm, BC = 8 cm, find AC.
- Using Pythagoras (because ∠B = 90°):
- AC² = AB² + BC² = 36 + 64 = 100
- AC = 10 cm
Example 4: Rhombus Diagonal
A rhombus has diagonals 12 cm and 16 cm. Find the side.
- Diagonals bisect at 90° in a rhombus
- Half-diagonals: 6 cm and 8 cm
- Side = √(6² + 8²) = √100 = 10 cm
Example 5: Trapezium Angles
In a trapezium ABCD, AB || CD, ∠A = 50°. Find ∠D.
- Co-interior angles (since AB || CD): ∠A + ∠D = 180°
- ∠D = 180° − 50° = 130°
Example 6: Identify Quadrilateral
ABCD has AB = CD = 5 cm, BC = AD = 5 cm, and ∠A = 90°. Identify it.
- All four sides equal → could be square or rhombus
- ∠A = 90° → square (in a rhombus, angles vary)
- Answer: Square
9. Common Mistakes
-
Confusing trapezium with parallelogram
- Trapezium: ONE pair parallel
- Parallelogram: BOTH pairs parallel
-
Forgetting that square is BOTH rectangle and rhombus
- A square has all properties of both
-
Wrong angle assumptions
- Don't assume angles are 90° unless told
- Don't assume sides are equal unless told
-
Diagonals confusion
- Parallelogram: bisect each other (but NOT equal, NOT perpendicular in general)
- Rectangle: equal + bisect (NOT perpendicular)
- Rhombus: perpendicular + bisect (NOT equal)
- Square: ALL properties
-
Construction without 5 measurements
- You need 5 measurements (in proper combinations) to UNIQUELY determine a quadrilateral
10. Special Properties Worth Remembering
Pythagoras in Quadrilaterals
Many problems use the Pythagoras theorem (which you'll formally study in Chapter 9):
- Rectangle diagonal: d = √(L² + B²)
- Square diagonal: d = a√2 (where a = side)
- Rhombus side from diagonals: s = √((d₁/2)² + (d₂/2)²)
Area Formulas (Preview for Chapter 14)
- Square: a²
- Rectangle: L × B
- Parallelogram: base × height
- Rhombus: (1/2) × d₁ × d₂
- Trapezium: (1/2) × (sum of parallel sides) × height
- Kite: (1/2) × d₁ × d₂
Perimeter Formulas
- Rectangle: 2(L + B)
- Square: 4a
- Rhombus: 4a (all sides equal)
- General quadrilateral: sum of all sides
11. Quadrilaterals Around You
Architecture
- Most rooms, walls, doors, windows are rectangles
- Floor tiles often square
- Some traditional motifs use rhombuses and kites
Art
- Cubist art (Picasso) uses quadrilateral shapes
- Indian temple architecture has many quadrilateral patterns
Nature
- Honeycomb cells are HEXAGONS, but some crystals are quadrilateral
- Many salt crystals are cubic (square faces)
Sports
- Cricket pitch, football field, hockey field — all rectangles
- Carom board — square
- Kabaddi court — rectangle
Vehicles
- Truck bodies, train coaches — rectangular sides
- Aircraft wings often trapezoidal
12. Tips for Mastery
For Identification
- Memorise the PROPERTIES of each type
- Use the FAMILY TREE to know inheritance (square = rectangle + rhombus)
For Properties Application
- Apply properties step by step
- Use the 360° rule for missing angles
For Construction
- Draw NEATLY using ruler, compass, protractor
- Use light pencil first; darken only after verifying
For Problem-Solving
- Identify the type of quadrilateral first
- Then apply specific properties
- Verify your answer (e.g., angle sum = 360°)
13. Historical Context
Ancient Indian Geometry
- Sulba Sutras had detailed quadrilateral constructions for fire altars
- Quadrilateral 'vedis' (altars) had specific dimensions
Greek Contribution
- Euclid (~300 BCE) systematised quadrilateral geometry in 'Elements'
- Theorems on parallelograms come from Euclid
Modern Use
- Quadrilateral mesh in computer graphics
- Image processing uses quadrilateral patches
- GPS triangulation uses quadrilateral coordinate systems
14. Conclusion
Quadrilaterals are everywhere — in architecture, design, art, and daily life. Mastering their properties is essential for:
- Later geometry (Class 9 onwards)
- Trigonometry (Class 10 onwards)
- Coordinate geometry (Class 9 onwards)
- 3D shapes (later chapters)
The single principle — angle sum = 360° — combined with the family-tree of types and their properties, gives you a complete toolkit for any quadrilateral problem.
Practice constructions with rigour, memorise properties precisely, and you'll find that the world around you is more geometric than you ever realised.
