Unit: Area Of Shapes
Chapter: Area of Rhombus
Reference: – What is a Rhombus, Area of Rhombus Using Diagonals (Formula and Derivation), Area of Rhombus Using Base and Height, Finding a Diagonal Given Area and Other Diagonal, Real-Life Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- How to Find the Area of a Rhombus Using Diagonals
- How to Find the Area of a Rhombus Using Base and Height
- How to Find a Missing Diagonal Given Area
- When to Use Each Formula
Introduction to Area of Rhombus
Definition
A rhombus is a parallelogram with all four sides equal. It looks like a slanted square or a diamond shape. Like other quadrilaterals, a rhombus encloses a certain amount of space called its area. Unlike a square, the angles of a rhombus are not necessarily 90°.
When we calculate the area of a rhombus, we essentially ask:
"How many square units fit inside this diamond shape?"
There are two different methods to find the area of a rhombus, and both are useful in different situations.
Importance of Area of Rhombus
- Used in kite making and diamond cutting
- Essential for tiling and flooring patterns
- Helps in solving geometry problems involving quadrilaterals
- Foundation for advanced geometry and trigonometry
Example
A rhombus with diagonals 6 cm and 8 cm has an area of 24 cm². The same rhombus has a base of 5 cm and height of 4.8 cm, giving the same area.
Subtopics
1. Properties of a Rhombus (Quick Review)
- All four sides are equal in length
- Opposite sides are parallel (it is a parallelogram)
- Opposite angles are equal
- Adjacent angles are supplementary (sum to 180°)
- Diagonals bisect each other at right angles (90°)
- Diagonals bisect the interior angles
2. Area of Rhombus – Method 1 (Using Diagonals)
This is the most common formula for the area of a rhombus.
Formula: Area = (1/2) × d₁ × d₂
Where d₁ and d₂ are the lengths of the two diagonals.
Derivation: The diagonals of a rhombus divide it into four congruent right triangles. Each triangle has legs equal to half of each diagonal (d₁/2 and d₂/2). The area of one triangle is (1/2) × (d₁/2) × (d₂/2) = d₁d₂/8. Multiplying by 4 triangles gives total area = 4 × (d₁d₂/8) = d₁d₂/2.
Example 1: A rhombus has diagonals 10 cm and 12 cm. Find its area.
Area = (1/2) × 10 × 12 = 60 cm²
Example 2: A rhombus has diagonals 14 m and 20 m. Find its area.
Area = (1/2) × 14 × 20 = 140 m²
Example 3: A rhombus has diagonals 9 cm and 16 cm. Find its area.
Area = (1/2) × 9 × 16 = 72 cm²
3. Area of Rhombus – Method 2 (Using Base and Height)
Since a rhombus is a parallelogram, we can also use the parallelogram area formula.
Formula: Area = base × height
Where base is the length of any side, and height is the perpendicular distance between that side and its opposite side.
Example 1: A rhombus has side length 8 cm and height 5 cm. Find its area.
Area = 8 × 5 = 40 cm²
Example 2: A rhombus has side length 12 m and height 7 m. Find its area.
Area = 12 × 7 = 84 m²
Note: The height is NOT the length of the other side. It is the perpendicular distance between parallel sides.
4. Finding a Diagonal Given Area
If the area and one diagonal are known, we can find the other diagonal.
Formula: d₂ = (2 × Area) ÷ d₁ (or d₁ = (2 × Area) ÷ d₂)
Example: A rhombus has area 80 cm² and one diagonal is 10 cm. Find the other diagonal.
d₂ = (2 × 80) ÷ 10 = 160 ÷ 10 = 16 cm
5. Finding Side Length from Diagonals
Since the diagonals are perpendicular and bisect each other, they form four right triangles. The side of the rhombus is the hypotenuse of a right triangle with legs equal to half of each diagonal.
Formula: side = √[(d₁/2)² + (d₂/2)²] = (1/2) × √(d₁² + d₂²)
Example: A rhombus has diagonals 6 cm and 8 cm. Find its side length.
Half diagonals = 3 cm and 4 cm
Side = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
Solved Examples
Example 1 – Area Using Diagonals: Find the area of a rhombus with diagonals 15 cm and 24 cm.
Solution: A = (1/2) × 15 × 24 = (1/2) × 360 = 180 cm²
Answer: 180 cm²
Example 2 – Area Using Base and Height: A rhombus has side length 9 m and height 6 m. Find its area.
Solution: A = base × height = 9 × 6 = 54 m²
Answer: 54 m²
Example 3 – Finding Missing Diagonal: A rhombus has area 120 cm² and one diagonal is 20 cm. Find the other diagonal.
Solution: d₂ = (2 × 120) ÷ 20 = 240 ÷ 20 = 12 cm
Answer: 12 cm
Example 4 – Finding Side from Diagonals: The diagonals of a rhombus are 10 cm and 24 cm. Find the side length.
Solution: Half diagonals = 5 cm and 12 cm
Side = √(5² + 12²) = √(25 + 144) = √169 = 13 cm
Answer: 13 cm
Common Mistakes to Avoid
Mistake 1 – Using side² for area of rhombus
Only a square has area = side². A non-square rhombus has less area than side².
Correct understanding: Use A = (1/2) × d₁ × d₂ or A = base × height.
Mistake 2 – Forgetting the 1/2 in diagonal formula
A = d₁ × d₂ (without 1/2) gives twice the actual area.
Correct understanding: Always include (1/2) or divide by 2.
Mistake 3 – Using slanted side as height
Height is the perpendicular distance between parallel sides, not the length of the slanted side.
Correct understanding: Draw a perpendicular line inside the shape to find height.
Mistake 4 – Confusing the diagonals
Both diagonals are used in the formula, not just one.
Correct understanding: Multiply both diagonals, then divide by 2.
Mistake 5 – Forgetting to take half of diagonals when finding side
Side = √[(d₁/2)² + (d₂/2)²], not √(d₁² + d₂²).
Correct understanding: Divide each diagonal by 2 before squaring.
Mistake 6 – Thinking all rhombuses are squares
A rhombus can have acute and obtuse angles (like a diamond).
Correct understanding: A square is a special type of rhombus, but not all rhombuses are squares.
Quick Reference Summary
Rhombus: Parallelogram with all sides equal
Area Formula (Diagonals): A = (1/2) × d₁ × d₂
Area Formula (Base & Height): A = b × h
To find missing diagonal: d₂ = (2 × A) ÷ d₁
To find side from diagonals: s = (1/2) × √(d₁² + d₂²)
Perimeter of Rhombus: P = 4s
Key Fact: In a rhombus, diagonals are perpendicular bisectors of each other
Square is a rhombus: When all angles = 90°, A = s² also = (1/2) × d²