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Quadrilaterals (Parallelogram, Rhombus, Trapezium)

Introduction

Quadrilaterals are four-sided polygons with diverse properties. In aptitude and geometry-based exams, problems often involve finding area, perimeter, and relations between sides, diagonals, or heights of figures like parallelograms, rhombuses, and trapeziums.

Understanding their unique formulas and geometric properties helps in quickly identifying the right approach during problem-solving.

Pattern: Quadrilaterals (Parallelogram, Rhombus, Trapezium)

Pattern

The key idea is to identify the type of quadrilateral and apply its respective area or diagonal formula efficiently.

Key Formulas:

  • Parallelogram: Area = base × height; Perimeter = 2(a + b)
  • Rhombus: Area = ½ × d₁ × d₂; All sides equal, diagonals bisect at right angles.
  • Trapezium: Area = ½ × (sum of parallel sides) × height

Step-by-Step Example

Question

The diagonals of a rhombus are 24 cm and 10 cm. Find its area.

Solution

  1. Step 1: Recall the formula for the area of a rhombus.

    Area = ½ × d₁ × d₂

  2. Step 2: Substitute the given values.

    Area = ½ × 24 × 10

  3. Step 3: Compute.

    Area = ½ × 240 = 120 cm²

  4. Final Answer:

    Area of the rhombus = 120 cm²

  5. Quick Check:

    Diagonals bisect at right angles; product of diagonals = 240 → half = 120 ✅

Quick Variations

1. Find base or height of a parallelogram when area and one dimension are given.

2. Calculate diagonal lengths of rhombus using Pythagoras theorem.

3. Use trapezium formula to find missing side or height.

4. Combined figure problems (e.g., parallelogram + triangle).

Trick to Always Use

  • Step 1 → Identify the shape first (parallelogram, rhombus, or trapezium).
  • Step 2 → Write down the formula linked with that shape.
  • Step 3 → Substitute the given values systematically - never skip the “½” for diagonals or trapeziums.

Summary

Summary

For Quadrilaterals (Parallelogram, Rhombus, Trapezium):

  • Identify the figure type first - formulas differ.
  • Parallelogram: base × height; Rhombus: ½ × d₁ × d₂; Trapezium: ½ × (a + b) × h.
  • Check units (cm², m²) after calculations.
  • Verify results by substituting back or comparing with approximate geometric shape.

Practice

(1/5)
1. Find the area of a parallelogram with base 12 cm and height 8 cm.
easy
A. 96 cm²
B. 100 cm²
C. 84 cm²
D. 90 cm²

Solution

  1. Step 1: Recall the formula for area of parallelogram.

    Area = base × height.

  2. Step 2: Substitute values.

    Area = 12 × 8 = 96 cm².

  3. Final Answer:

    Area = 96 cm² → Option A.

  4. Quick Check:

    12 × 8 = 96 ✅
Hint: Multiply base by height directly.
Common Mistakes: Using ½ × base × height (that’s for triangles).
2. The diagonals of a rhombus are 16 cm and 12 cm. Find its area.
easy
A. 96 cm²
B. 90 cm²
C. 100 cm²
D. 80 cm²

Solution

  1. Step 1: Use formula for area of rhombus.

    Area = ½ × d₁ × d₂.

  2. Step 2: Substitute values.

    Area = ½ × 16 × 12 = ½ × 192 = 96 cm².

  3. Final Answer:

    Area = 96 cm² → Option A.

  4. Quick Check:

    Half of 192 = 96 ✅
Hint: Multiply diagonals and take half.
Common Mistakes: Forgetting the ½ in the formula.
3. A trapezium has parallel sides of 10 cm and 6 cm, and height 4 cm. Find its area.
easy
A. 30 cm²
B. 32 cm²
C. 36 cm²
D. 34 cm²

Solution

  1. Step 1: Use formula for trapezium area.

    Area = ½ × (sum of parallel sides) × height.

  2. Step 2: Substitute values.

    Area = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 8 × 4 = 32 cm².

  3. Final Answer:

    Area = 32 cm² → Option B.

  4. Quick Check:

    (10 + 6)/2 = 8 × 4 = 32 ✅
Hint: Take average of parallel sides and multiply by height.
Common Mistakes: Adding height to sides instead of multiplying.
4. If the area of a parallelogram is 150 cm² and its base is 15 cm, find its height.
medium
A. 8 cm
B. 12 cm
C. 14 cm
D. 10 cm

Solution

  1. Step 1: Recall area formula.

    Area = base × height.

  2. Step 2: Substitute and rearrange.

    150 = 15 × height → height = 150 ÷ 15 = 10 cm.

  3. Final Answer:

    Height = 10 cm → Option D.

  4. Quick Check:

    15 × 10 = 150 ✅
Hint: Divide area by base to find height.
Common Mistakes: Multiplying instead of dividing.
5. In a rhombus, each side is 13 cm and one diagonal is 10 cm. Find the other diagonal.
medium
A. 20 cm
B. 22 cm
C. 24 cm
D. 18 cm

Solution

  1. Step 1: Use property of rhombus diagonals.

    Diagonals bisect each other at right angles.

  2. Step 2: Apply Pythagoras theorem.

    (d₁/2)² + (d₂/2)² = side² → (10/2)² + (d₂/2)² = 13².

  3. Step 3: Simplify.

    25 + (d₂/2)² = 169 → (d₂/2)² = 144 → d₂/2 = 12 → d₂ = 24 cm.

  4. Final Answer:

    Other diagonal = 24 cm → Option C.

  5. Quick Check:

    5² + 12² = 13² ✅
Hint: Use diagonals’ half-lengths with Pythagoras theorem.
Common Mistakes: Forgetting to divide diagonals by 2 before applying the theorem.

Mock Test

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