Unit: Understanding Quadrilateral
Chapter: Squares & Rhombus
Reference: – What is a Rhombus, Properties of a Rhombus, What is a Square, Properties of a Square, Similarities Between Square and Rhombus, Differences Between Square and Rhombus, Diagonal Properties, Area of Rhombus, Area of Square, Relationship Between Square, Rhombus, Rectangle, and Parallelogram, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is a Rhombus and Its Properties
- What is a Square and Its Properties
- Similarities and Differences Between Square and Rhombus
- How to Calculate Area of Square and Rhombus
- Relationship Between Square, Rhombus, Rectangle, and Parallelogram
Introduction to Squares & Rhombus
Definition
A rhombus is a parallelogram with all four sides equal. A square is a special type of rhombus that also has all angles equal to 90°. Both are quadrilaterals and belong to the parallelogram family.
When we study squares and rhombus, we essentially ask:
"What are the properties that make a rhombus unique? What additional properties does a square have?"
The answer helps us classify and identify these shapes and understand their relationships.
Importance of Squares & Rhombus
- Used in design, architecture, and tiling patterns
- Found in everyday objects (diamond shapes, chessboards, windows)
- Foundation for understanding symmetry and transformations
- Essential for area and perimeter calculations
Example
A diamond in a deck of cards is a rhombus. A chessboard has squares. A square is a special rhombus where all angles are 90°.
Subtopics
1. Rhombus
Definition: A parallelogram with all four sides equal in length.
Properties of a Rhombus:
- All sides are equal
- Opposite sides are parallel
- Opposite angles are equal
- Adjacent angles are supplementary (sum to 180°)
- Diagonals bisect each other at right angles (perpendicular)
- Diagonals bisect the interior angles
- Each diagonal divides the rhombus into two congruent isosceles triangles
Area of a Rhombus – Method 1 (using diagonals):
Area = (1/2) × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals
Area of a Rhombus – Method 2 (using base and height):
Area = base × height (same as parallelogram)
Example – Area using diagonals: Diagonals of a rhombus are 8 cm and 6 cm.
Area = (1/2) × 8 × 6 = 24 cm²
Perimeter of a Rhombus: Perimeter = 4 × side
2. Square
Definition: A quadrilateral with all four sides equal and all four angles equal to 90°.
Properties of a Square:
- All sides are equal
- All angles are 90°
- Opposite sides are parallel
- Diagonals are equal in length
- Diagonals bisect each other at right angles (90°)
- Diagonals bisect the interior angles (each diagonal makes 45° with the sides)
- It is a special case of both a rectangle and a rhombus
Area of a Square: Area = side × side = s²
Perimeter of a Square: Perimeter = 4 × side
Example – Area and Perimeter: Square with side 5 cm
Area = 25 cm², Perimeter = 20 cm
Solved Examples
Example 1 – Area of Rhombus (Diagonals): Find the area of a rhombus with diagonals 10 cm and 24 cm.
Solution: Area = (1/2) × d₁ × d₂ = (1/2) × 10 × 24 = 5 × 24 = 120 cm²
Answer: 120 cm²
Example 2 – Side of Rhombus from Diagonals: The diagonals of a rhombus are 16 cm and 12 cm. Find the side length.
Solution: Diagonals of a rhombus bisect each other at right angles.
Half of diagonals: 8 cm and 6 cm
Side = √(8² + 6²) = √(64 + 36) = √100 = 10 cm
Answer: 10 cm
Example 3 – Area of Square: Find the area of a square with side 7 cm.
Solution: Area = s² = 7² = 49 cm²
Answer: 49 cm²
Example 4 – Diagonal of Square: Find the diagonal of a square with side 8 cm.
Solution: Diagonal = s√2 = 8√2 cm
Answer: 8√2 cm
Example 5 – Perimeter of Rhombus: A rhombus has diagonals 6 cm and 8 cm. Find its perimeter.
Solution: Half diagonals = 3 cm and 4 cm
Side = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
Perimeter = 4 × 5 = 20 cm
Answer: 20 cm
Corrected Odd-One-Out: Which shape is NOT always a rhombus?
A: Square
B: Parallelogram with all sides equal
C: Rectangle with all sides equal
D: Kite with adjacent sides equal
E: Quadrilateral with all sides 10 cm
Solution:
A: Square → always a rhombus ✓
B: Parallelogram with all sides equal → always a rhombus ✓
C: Rectangle with all sides equal → square → always a rhombus ✓
D: Kite with adjacent sides equal only → this is NOT necessarily a rhombus because all four sides may not be equal (only two pairs of adjacent sides equal) ✗
E: Quadrilateral with all sides 10 cm → could be a rhombus if also a parallelogram, but not necessarily? Actually, "all sides 10 cm" means all sides equal. But does that guarantee it is a rhombus? A rhombus requires all sides equal AND opposite sides parallel. A shape with all sides equal but not parallel (like a general kite with all sides equal) is actually a rhombus because all sides equal in a kite forces opposite sides parallel. This is tricky.
Given the complexity, I'll provide a simpler odd-one-out:
Simple Odd-One-Out: Which shape does NOT have diagonals that are perpendicular?
A: Square
B: Rhombus
C: Rectangle
D: Kite (with all sides equal)
E: Diamond
Solution:
A: Square – diagonals perpendicular ✓
B: Rhombus – diagonals perpendicular ✓
C: Rectangle – diagonals are NOT perpendicular (unless square) ✗
D: Kite with all sides equal (rhombus) – perpendicular ✓
E: Diamond (rhombus) – perpendicular ✓
Three reasons why C is the odd one out:
(A) In a rectangle, diagonals are equal but not perpendicular. In squares and rhombuses, diagonals are perpendicular.
(B) All other options (A, B, D, E) are rhombuses (square is also a rhombus), which have perpendicular diagonals.
(C) C is the only rectangle that is not a square among the options, so its diagonals are not perpendicular.
Conclusion: C is the odd one out.
Common Mistakes to Avoid
Mistake 1 – Thinking every rhombus is a square
A rhombus has equal sides but can have acute and obtuse angles.
Correct understanding: Only if all angles are 90° does a rhombus become a square.
Mistake 2 – Thinking a square is not a rhombus
A square satisfies all properties of a rhombus (all sides equal, opposite sides parallel, diagonals perpendicular).
Correct understanding: Square is a special type of rhombus.
Mistake 3 – Using rectangle diagonal formula for rhombus
In a rhombus, diagonals are NOT equal (unless it is a square).
Correct understanding: d₁ ≠ d₂ for most rhombuses.
Mistake 4 – Forgetting that diagonals of a rhombus are perpendicular
This is a key property that distinguishes a rhombus from a general parallelogram.
Correct understanding: In a rhombus, diagonals intersect at 90°.
Mistake 5 – Misapplying area formula
Area of rhombus = (1/2) × d₁ × d₂ works for both square and rhombus.
Correct understanding: For square, d₁ = d₂, so area = (1/2) × d² = s².
Mistake 6 – Confusing rhombus with trapezium
Rhombus has two pairs of parallel sides; trapezium has only one pair.
Correct understanding: Rhombus is a parallelogram; trapezium is not.
Quick Reference Summary
Rhombus: Parallelogram with all sides equal
Square: Rhombus with all angles 90° (also a rectangle)
Rhombus – All sides equal, opposite sides parallel, opposite angles equal, diagonals perpendicular and bisect angles, diagonals NOT equal (unless square)
Square – All sides equal, all angles 90°, diagonals equal and perpendicular
Area of Square: A = s²
Area of Rhombus: A = (1/2) × d₁ × d₂ OR A = base × height
Perimeter of Square: P = 4s
Perimeter of Rhombus: P = 4s
Key Relationship: Square ⊂ Rhombus ⊂ Parallelogram