8. Rhombus Formulas
A rhombus is a quadrilateral in which all four sides are equal.
In the rhombus diagram, each side is represented by a, while the two diagonals are represented by d₁ and d₂.
The important rhombus formulas given in the formula sheet are related to its diagonals, area, perimeter and diagonal relation.
What Is a Rhombus?
d₁ and d₂ are the two diagonals, and a is the length of each side.
1. Diagonals of a Rhombus
The diagonals of a rhombus bisect each other at 90°.
This means that the two diagonals intersect at right angles and each diagonal divides the other into two equal parts.
If the diagonals intersect at point O, then:
and the angle between the diagonals is:
2. Area of a Rhombus
The area of a rhombus can be found using its two diagonals.
Where:
- d₁ = First diagonal
- d₂ = Second diagonal
Example 1: Find the Area
A rhombus has diagonals of length 12 cm and 8 cm. Find its area.
= ½ × 12 × 8
= 6 × 8
= 48 cm²
Answer: 48 cm²
Example 2: Find a Missing Diagonal
The area of a rhombus is 60 cm² and one diagonal is 10 cm. Find the other diagonal.
Using:
Therefore:
d₂ = (2 × 60) ÷ 10
d₂ = 12 cm
Answer: 12 cm
3. Perimeter of a Rhombus
All four sides of a rhombus are equal. If each side has length a, then:
Example 1
Each side of a rhombus is 9 cm. Find its perimeter.
P = 4 × 9
P = 36 cm
Answer: 36 cm
Example 2: Find the Side
The perimeter of a rhombus is 52 cm. Find the length of each side.
a = 52 ÷ 4
a = 13 cm
Answer: 13 cm
4. Diagonal Relation of a Rhombus
The PDF gives an important relationship between the two diagonals and the side of a rhombus:
Where:
- d₁ = First diagonal
- d₂ = Second diagonal
- a = Side of the rhombus
Why Does This Relation Work?
Since the diagonals of a rhombus bisect each other at 90°, half of each diagonal and one side form a right-angled triangle.
Using the Pythagorean theorem:
Multiplying by 4:
Example 1
The diagonals of a rhombus are 6 cm and 8 cm. Find its side.
6² + 8² = 4a²
36 + 64 = 4a²
100 = 4a²
a² = 25
a = 5 cm
Answer: 5 cm
Example 2: Verify the Diagonal Relation
A rhombus has side 5 cm and diagonals 6 cm and 8 cm. Verify the formula.
Left side:
= 36 + 64
= 100
Right side:
= 4 × 25
= 100
Therefore:
Hence, the relation is verified.
5. Finding the Side Using the Diagonals
From:
we can rearrange the formula to find the side:
Example
The diagonals of a rhombus are 10 cm and 24 cm. Find the side.
= ½√(10² + 24²)
= ½√(100 + 576)
= ½√676
= ½ × 26
a = 13 cm
Answer: 13 cm
6. Area and Perimeter of a Rhombus
| Quantity | Formula | Unit |
|---|---|---|
| Area | ½ × d₁ × d₂ | Square units |
| Perimeter | 4a | Length units |
| Diagonal Relation | d₁² + d₂² = 4a² | Squared units |
Area uses the two diagonals.
Perimeter uses the side.
Diagonal relation connects the two diagonals with the side.
7. Combined Exam Example
A rhombus has diagonals of 12 cm and 16 cm. Find:
- Its area
- Its side
- Its perimeter
Step 1: Area
= ½ × 12 × 16
A = 96 cm²
Step 2: Side
= ½√(12² + 16²)
= ½√(144 + 256)
= ½√400
a = 10 cm
Step 3: Perimeter
= 4 × 10
P = 40 cm
Final Answers:
- Area = 96 cm²
- Side = 10 cm
- Perimeter = 40 cm
8. Common Mistakes in Rhombus Questions
Mistake 1: Forgetting that all sides are equal
For a rhombus:
Therefore:
Mistake 2: Forgetting the ½ in the area formula
The correct formula is:
Mistake 3: Forgetting that diagonals are perpendicular
The diagonals of a rhombus bisect each other at 90°.
Mistake 4: Using the parallelogram diagonal relation
For a rhombus, the source gives:
Do not confuse it with the general parallelogram relation.
9. Rhombus Formula Chart
| Concept | Formula / Property |
|---|---|
| Diagonal Property | Diagonals bisect each other at 90° |
| Area | ½ × d₁ × d₂ |
| Perimeter | 4a |
| Diagonal Relation | d₁² + d₂² = 4a² |
| Side from Diagonals | ½√(d₁² + d₂²) |
10. Practice Questions
Question 1: A rhombus has diagonals 14 cm and 10 cm. Find its area.
Question 2: Each side of a rhombus is 15 cm. Find its perimeter.
Question 3: The diagonals of a rhombus are 6 cm and 8 cm. Find its side.
Question 4: The area of a rhombus is 120 cm² and one diagonal is 12 cm. Find the other diagonal.
Question 5: The diagonals of a rhombus are 10 cm and 24 cm. Find its side and perimeter.
Question 6: A rhombus has side 13 cm and one diagonal 10 cm. Using the diagonal relation, find the other diagonal.
11. Rhombus Formula Quick Revision
12. Mensuration Exam Tip
When a rhombus question gives the two diagonals, think of the area formula first:
When the question asks for the side from the diagonals, use:
When the question asks for the perimeter, use:
In Part 9 — Trapezium Formulas, we will cover perimeter, area, height, diagonals and the important diagonal relation of a trapezium, with diagrams, explanations, solved examples, practice questions and exam tips.