Solved Exercises on Types of Quadrilaterals and Their Properties in Grade 8

Master quadrilaterals: squares, rectangles, rhombuses, parallelograms, trapezoids with these 5 detailed exercises covering all properties and applications.

Types of Quadrilaterals and Their Properties
1 Square Properties
Exercise 1
A square has sides of length 8 cm. Find its perimeter, area, and diagonal length.
Definition:

Square: A quadrilateral with all sides equal and all angles 90°

Method:
  1. Identify the side length (s = 8 cm)
  2. Apply square formulas: P = 4s, A = s², d = s√2
  3. Calculate each property
SQUARE
  • ✅ All 4 sides equal
  • ✅ All 4 angles 90°
  • ✅ Diagonals equal and bisect at 90°
  • ✅ Opposite sides parallel
    • Step 1: Identify given information

    Side length (s) = 8 cm

    Step 2: Calculate perimeter

    Perimeter = 4 × side = 4 × 8 = 32 cm

    Step 3: Calculate area

    Area = side² = 8² = 64 cm²

    Step 4: Calculate diagonal

    Diagonal = side × √2 = 8 × √2 ≈ 8 × 1.414 = 11.31 cm

    Perimeter = 32 cm
    Area = 64 cm²
    Diagonal ≈ 11.31 cm
    Final answer:

    Perimeter = 32 cm, Area = 64 cm², Diagonal ≈ 11.31 cm

    Applied formulas:

    • Perimeter: P = 4s

    • Area: A = s²

    • Diagonal: d = s√2

    2 Rectangle Properties
    Exercise 2
    A rectangle has length 12 cm and width 5 cm. Find its perimeter, area, and diagonal length.
    Definition:

    Rectangle: A quadrilateral with opposite sides equal and all angles 90°

    RECTANGLE
  • ✅ Opposite sides equal and parallel
  • ✅ All 4 angles 90°
  • ✅ Diagonals equal and bisect each other
  • ✅ Adjacent angles supplementary
    • Step 1: Identify given information

    Length (l) = 12 cm, Width (w) = 5 cm

    Step 2: Calculate perimeter

    Perimeter = 2(l + w) = 2(12 + 5) = 2 × 17 = 34 cm

    Step 3: Calculate area

    Area = l × w = 12 × 5 = 60 cm²

    Step 4: Calculate diagonal

    Diagonal = √(l² + w²) = √(12² + 5²) = √(144 + 25) = √169 = 13 cm

    Perimeter = 34 cm
    Area = 60 cm²
    Diagonal = 13 cm
    Final answer:

    Perimeter = 34 cm, Area = 60 cm², Diagonal = 13 cm

    Applied formulas:

    • Perimeter: P = 2(l + w)

    • Area: A = l × w

    • Diagonal: d = √(l² + w²)

    3 Rhombus Properties
    Exercise 3
    A rhombus has diagonals of lengths 10 cm and 24 cm. Find its area, perimeter, and side length.
    Definition:

    Rhombus: A quadrilateral with all sides equal and diagonals perpendicular

    RHOMBUS
  • ✅ All 4 sides equal
  • ✅ Opposite sides parallel
  • ✅ Diagonals perpendicular and bisect each other
  • ✅ Opposite angles equal
    • Step 1: Identify given information

    Diagonal 1 (d₁) = 10 cm, Diagonal 2 (d₂) = 24 cm

    Step 2: Calculate area

    Area = (d₁ × d₂)/2 = (10 × 24)/2 = 240/2 = 120 cm²

    Step 3: Calculate side length

    Since diagonals bisect at right angles, side = √[(d₁/2)² + (d₂/2)²]

    Side = √[(10/2)² + (24/2)²] = √[5² + 12²] = √[25 + 144] = √169 = 13 cm

    Step 4: Calculate perimeter

    Perimeter = 4 × side = 4 × 13 = 52 cm

    Area = 120 cm²
    Side length = 13 cm
    Perimeter = 52 cm
    Final answer:

    Area = 120 cm², Side length = 13 cm, Perimeter = 52 cm

    Applied formulas:

    • Area: A = (d₁ × d₂)/2

    • Side: s = √[(d₁/2)² + (d₂/2)²]

    • Perimeter: P = 4s

    Quadrilateral Types Overview
    🟥
    Square
    All sides equal, all angles 90°
    🟦
    Rectangle
    Opposite sides equal, all angles 90°
    🔷
    Rhombus
    All sides equal, diagonals ⊥
    ▰
    Parallelogram
    Opposite sides parallel and equal
    梯
    Trapezoid
    One pair of parallel sides
    \(\text{Sum of interior angles} = 360°\)
    Universal Property
    Key definitions:

    Quadrilateral: A polygon with 4 sides and 4 vertices

    Convex: All interior angles less than 180°

    Concave: At least one interior angle greater than 180°

    Common properties: 4 sides, 4 vertices, 4 interior angles summing to 360°
    Hierarchy: Square ⊂ Rectangle ⊂ Parallelogram ⊂ Trapezoid ⊂ Quadrilateral
    Additional Exercises
    4 Parallelogram Properties
    Exercise 4
    In a parallelogram ABCD, angle A = 70°. Find all other angles.
    Definition:

    Parallelogram: A quadrilateral with opposite sides parallel and equal

    PARALLELOGRAM
  • ✅ Opposite sides parallel and equal
  • ✅ Opposite angles equal
  • ✅ Consecutive angles supplementary
  • ✅ Diagonals bisect each other
    • Step 1: Apply parallelogram angle properties

    • Opposite angles are equal: ∠A = ∠C = 70°

    • Consecutive angles are supplementary: ∠A + ∠B = 180°

    Step 2: Calculate ∠B

    ∠B = 180° - ∠A = 180° - 70° = 110°

    Step 3: Find ∠D

    ∠D = ∠B (opposite angles) = 110°

    ∠A = 70°
    ∠B = 110°
    ∠C = 70°
    ∠D = 110°
    Final answer:

    ∠A = 70°, ∠B = 110°, ∠C = 70°, ∠D = 110°

    Applied properties:

    • Opposite angles equal: ∠A = ∠C, ∠B = ∠D

    • Consecutive angles supplementary: ∠A + ∠B = 180°

    • Sum of all angles: 360°

    5 Trapezoid Properties
    Exercise 5
    A trapezoid has parallel sides of 10 cm and 6 cm, and height 4 cm. Find its area.
    Definition:

    Trapezoid: A quadrilateral with exactly one pair of parallel sides

    TRAPEZOID
  • ✅ One pair of parallel sides (bases)
  • ✅ Legs may or may not be equal
  • ✅ Base angles may or may not be equal
  • ✅ Height is perpendicular distance between bases
    • Step 1: Identify given information

    Base 1 (b₁) = 10 cm, Base 2 (b₂) = 6 cm, Height (h) = 4 cm

    Step 2: Apply trapezoid area formula

    Area = (1/2) × (b₁ + b₂) × h

    Step 3: Calculate area

    Area = (1/2) × (10 + 6) × 4 = (1/2) × 16 × 4 = 8 × 4 = 32 cm²

    Step 4: Verify calculation

    (1/2) × (10 + 6) × 4 = (1/2) × 16 × 4 = 32 cm² ✓

    Area = 32 cm²
    Final answer:

    Area = 32 cm²

    Applied formulas:

    • Area: A = (1/2) × (b₁ + b₂) × h

    • Parallel sides: Called bases

    • Height: Perpendicular distance between bases

    Comprehensive Properties Guide
    \(\text{Interior Angle Sum} = (n-2) \times 180° = 360°\)
    General Formula (n=4)
    Key definitions:

    Quadrilateral: Polygon with 4 sides, 4 vertices, 4 angles

    Adjacent sides: Two sides sharing a common vertex

    Opposite sides: Sides not sharing a vertex

    Diagonal: Line segment connecting non-adjacent vertices

    Identification method:
    1. Count sides: Must have exactly 4 sides
    2. Check angle sum: Must total 360°
    3. Identify special properties: Equal sides, parallel sides, right angles
    4. Classify: Based on specific properties
    Tip 1: Every square is also a rectangle, rhombus, and parallelogram.
    Tip 2: Diagonals of a rectangle are equal and bisect each other.
    Tip 3: Diagonals of a rhombus are perpendicular and bisect each other.
    Tip 4: In a parallelogram, opposite sides and angles are equal.
    Common errors: Confusing properties between different quadrilaterals, forgetting that squares satisfy all rectangle and rhombus properties.
    Exam preparation: Memorize formulas for area and perimeter, understand property hierarchies, practice angle calculations.
    Essential formulas:

    • Square: P = 4s, A = s², d = s√2

    • Rectangle: P = 2(l+w), A = lw, d = √(l²+w²)

    • Rhombus: P = 4s, A = (d₁d₂)/2

    • Parallelogram: P = 2(a+b), A = bh

    • Trapezoid: A = (1/2)(b₁+b₂)h

    Visual Properties Comparison
    Property Square Rectangle Rhombus Parallelogram Trapezoid
    All sides equal ✓ ✗ ✓ ✗ ✗
    Opposite sides equal ✓ ✓ ✓ ✓ ✗
    All angles 90° ✓ ✓ ✗ ✗ ✗
    Opposite angles equal ✓ ✓ ✓ ✓ ✗
    Diagonals equal ✓ ✓ ✗ ✗ ✗
    Diagonals ⊥ ✓ ✗ ✓ ✗ ✗

    Key Relationships:

    • Square ⊂ Rectangle ⊂ Parallelogram ⊂ Trapezoid ⊂ Quadrilateral
    • Square ⊂ Rhombus ⊂ Parallelogram ⊂ Trapezoid ⊂ Quadrilateral
    • Every square is simultaneously a rectangle, rhombus, and parallelogram

    Questions & Answers

    Question: I'm confused about the relationship between squares, rectangles, and rhombuses. Is a square a rectangle? Is it also a rhombus? How do I remember these relationships?

    Answer: Great question! Yes, a square is both a rectangle AND a rhombus. Think of it as a special case that satisfies ALL properties:

    • As a rectangle: It has all angles equal to 90°
    • As a rhombus: It has all sides equal
    • As a parallelogram: It has opposite sides parallel and equal

    Think of it as a hierarchy: Square ⊂ Rectangle ⊂ Parallelogram ⊂ Trapezoid ⊂ Quadrilateral

    Every square is automatically a rectangle (because it has 4 right angles), but not every rectangle is a square. Similarly, every square is a rhombus (because it has 4 equal sides), but not every rhombus is a square.

    Question: How do I find the area of a rhombus if I only know the side length? Do I need the diagonals?

    Answer: There are two ways to find the area of a rhombus:

    • If you know the diagonals: Area = (d₁ × d₂)/2
    • If you know side length and an angle: Area = s² × sin(angle)
    • If you know side length and height: Area = base × height

    However, knowing just the side length alone is not sufficient to determine the area, because the area depends on how "squashed" the rhombus is. You need either the diagonals, an angle, or the height (perpendicular distance between parallel sides).

    The diagonal method is usually preferred when available because it's straightforward: multiply the diagonals and divide by 2.

    Question: In a parallelogram, if one angle is 60°, what are the measures of all other angles?

    Answer: In a parallelogram, you can find all angles using two key properties:

    • Opposite angles are equal
    • Consecutive angles are supplementary (add up to 180°)

    If one angle is 60°:

    • The opposite angle is also 60°
    • The consecutive angles are 180° - 60° = 120°
    • The fourth angle (opposite to the 120° angle) is also 120°

    So the four angles are: 60°, 120°, 60°, 120°

    Always check: 60° + 120° + 60° + 120° = 360° ✓

    Question: What's the difference between a trapezoid and a trapezium? Are they the same thing?

    Answer: This is a common source of confusion due to regional differences:

    • In American English: A trapezoid has exactly ONE pair of parallel sides
    • In British English: A trapezium has exactly ONE pair of parallel sides
    • In American English: A trapezium has NO parallel sides
    • In British English: A trapezoid has NO parallel sides

    For your grade 8 curriculum, you'll typically encounter the American definition where a trapezoid has exactly one pair of parallel sides. The important thing is that in your textbook, a trapezoid has exactly one pair of parallel sides, which distinguishes it from parallelograms that have two pairs of parallel sides.

    Question: Is a kite a type of quadrilateral we need to study? How does it relate to other quadrilaterals?

    Answer: Yes, kites are important quadrilaterals! A kite has two pairs of adjacent sides equal, but opposite sides are not necessarily equal. Key properties:

    • Two pairs of adjacent sides equal
    • One pair of opposite angles equal
    • Diagonals are perpendicular
    • One diagonal bisects the other

    Relationships: Every square is a kite (since it has two pairs of adjacent equal sides), but not every kite is a square. A rhombus is a special type of kite where all four sides are equal. Kites are distinct from parallelograms because opposite sides are not necessarily parallel.

    For your grade 8 studies, focus on the main types (square, rectangle, rhombus, parallelogram, trapezoid), but understanding kites helps with the complete picture of quadrilateral classification.