Area of Rhombus
A rhombus is a parallelogram in which all four sides are equal. Finding the area of a rhombus is an important topic in Class 8 Mensuration. Unlike a rectangle (where area = length x breadth), the area of a rhombus is most commonly calculated using its diagonals.
The formula for the area of a rhombus is: Area = (1/2) x d₁ x d₂, where d₁ and d₂ are the lengths of the two diagonals. This formula works because the diagonals of a rhombus bisect each other at right angles, dividing the rhombus into four congruent right triangles.
In this topic, you will learn the formula, understand why it works, and solve a variety of problems involving the area of a rhombus — from straightforward calculations to word problems involving real-life applications.
What is Area of Rhombus?
Definition: The area of a rhombus is the amount of space enclosed within its four sides. It is measured in square units (cm², m², etc.).
Two methods to find the area:
- Method 1 (Using diagonals): Area = (1/2) x d₁ x d₂
- Method 2 (Using base and height): Area = base x height (since a rhombus is a parallelogram)
Method 1 is more commonly used because the diagonals of a rhombus are easier to measure or calculate than the height.
Area of Rhombus Formula
Primary Formula:
Area = (1/2) x d₁ x d₂
Where:
- d₁ = length of the first diagonal
- d₂ = length of the second diagonal
Alternative Formula (as a parallelogram):
Area = base x height = side x h
Where h is the perpendicular distance between two parallel sides.
Related formulas:
- Diagonal from area: d₂ = 2 x Area / d₁
- Side from diagonals: side = sqrt[(d₁/2)² + (d₂/2)²]
- Perimeter: P = 4 x side
Derivation and Proof
Why Area = (1/2) x d₁ x d₂?
The diagonals of a rhombus bisect each other at right angles. Let the diagonals be AC = d₁ and BD = d₂, intersecting at point O.
Step 1: Since the diagonals bisect each other:
- OA = OC = d₁/2
- OB = OD = d₂/2
Step 2: The diagonals divide the rhombus into 4 right triangles: AOB, BOC, COD, and DOA. Each is a right triangle (the right angle is at O).
Step 3: Area of each right triangle:
- Area of triangle AOB = (1/2) x OA x OB = (1/2) x (d₁/2) x (d₂/2)
Step 4: All 4 triangles are congruent (same legs), so:
- Total area = 4 x (1/2) x (d₁/2) x (d₂/2)
- = 4 x (d₁ x d₂) / 8
- = d₁ x d₂ / 2
- = (1/2) x d₁ x d₂
Alternative derivation:
The rhombus can be seen as two congruent triangles formed by one diagonal. If the diagonal AC = d₁ is the base, then the two triangles have heights OB = d₂/2 and OD = d₂/2.
- Area of upper triangle = (1/2) x d₁ x (d₂/2)
- Area of lower triangle = (1/2) x d₁ x (d₂/2)
- Total = 2 x (1/2) x d₁ x (d₂/2) = (1/2) x d₁ x d₂
Types and Properties
Area of rhombus problems can be classified as follows:
1. Direct calculation — both diagonals given:
- Apply Area = (1/2) x d₁ x d₂ directly.
2. Finding a diagonal — area and one diagonal given:
- Rearrange: d₂ = 2 x Area / d₁.
3. Finding area — side and one diagonal given:
- Use Pythagoras theorem to find the other diagonal first.
- side² = (d₁/2)² + (d₂/2)². Solve for d₂, then find area.
4. Finding area — side and height given:
- Use Area = side x height (parallelogram formula).
5. Comparing areas:
- Compare the area of a rhombus with another shape (rectangle, square, triangle).
6. Word problems:
- Real-life problems involving rhombus-shaped fields, tiles, windows, etc.
7. Finding area when perimeter and diagonal are given:
- First find the side from perimeter (side = P/4), then use Pythagoras to find the other diagonal, then calculate area.
Solved Examples
Example 1: Example 1: Area from two diagonals
Problem: The diagonals of a rhombus are 18 cm and 14 cm. Find the area.
Solution:
Given:
- d₁ = 18 cm, d₂ = 14 cm
Using the formula:
- Area = (1/2) x d₁ x d₂
- = (1/2) x 18 x 14
- = (1/2) x 252
- = 126 cm²
Answer: The area is 126 cm².
Example 2: Example 2: Finding the other diagonal
Problem: The area of a rhombus is 240 cm² and one diagonal is 20 cm. Find the other diagonal.
Solution:
Given:
- Area = 240 cm², d₁ = 20 cm
Using the formula:
- 240 = (1/2) x 20 x d₂
- 240 = 10 x d₂
- d₂ = 240/10 = 24 cm
Answer: The other diagonal is 24 cm.
Example 3: Example 3: Area using side and diagonal
Problem: A rhombus has a side of 10 cm and one diagonal of 12 cm. Find the area.
Solution:
Given:
- Side = 10 cm, d₁ = 12 cm
Finding d₂:
- side² = (d₁/2)² + (d₂/2)²
- 10² = 6² + (d₂/2)²
- 100 = 36 + (d₂/2)²
- (d₂/2)² = 64
- d₂/2 = 8
- d₂ = 16 cm
Finding area:
- Area = (1/2) x 12 x 16 = 96 cm²
Answer: The area is 96 cm².
Example 4: Example 4: Area using base and height
Problem: A rhombus has a side of 15 cm and a height of 8 cm. Find the area.
Solution:
Given:
- Side (base) = 15 cm, height = 8 cm
Since a rhombus is a parallelogram:
- Area = base x height = 15 x 8 = 120 cm²
Answer: The area is 120 cm².
Example 5: Example 5: Finding the side from area and diagonals
Problem: A rhombus has diagonals 6 cm and 8 cm. Find the side and the perimeter.
Solution:
Given:
- d₁ = 6 cm, d₂ = 8 cm
Finding the side:
- side² = (3)² + (4)² = 9 + 16 = 25
- side = 5 cm
Finding perimeter:
- Perimeter = 4 x 5 = 20 cm
Finding area:
- Area = (1/2) x 6 x 8 = 24 cm²
Answer: Side = 5 cm, Perimeter = 20 cm, Area = 24 cm².
Example 6: Example 6: Perimeter and diagonal given
Problem: The perimeter of a rhombus is 52 cm and one diagonal is 10 cm. Find the area.
Solution:
Given:
- Perimeter = 52 cm, so side = 52/4 = 13 cm
- d₁ = 10 cm, so d₁/2 = 5 cm
Finding d₂:
- side² = (d₁/2)² + (d₂/2)²
- 13² = 5² + (d₂/2)²
- 169 = 25 + (d₂/2)²
- (d₂/2)² = 144
- d₂/2 = 12
- d₂ = 24 cm
Finding area:
- Area = (1/2) x 10 x 24 = 120 cm²
Answer: The area is 120 cm².
Example 7: Example 7: Rhombus-shaped tile
Problem: A floor tile is in the shape of a rhombus with diagonals 8 cm and 6 cm. How many tiles are needed to cover a floor area of 9600 cm²?
Solution:
Area of one tile:
- = (1/2) x 8 x 6 = 24 cm²
Number of tiles:
- = 9600 / 24 = 400
Answer: 400 tiles are needed.
Example 8: Example 8: Comparing with rectangle area
Problem: A rhombus and a rectangle have the same base of 12 cm. The rhombus has a height of 7 cm and the rectangle has a height of 9 cm. Find which has a greater area.
Solution:
Rhombus area:
- Area = base x height = 12 x 7 = 84 cm²
Rectangle area:
- Area = length x breadth = 12 x 9 = 108 cm²
Answer: The rectangle has a greater area (108 > 84).
Example 9: Example 9: Area when diagonals are in a ratio
Problem: The diagonals of a rhombus are in the ratio 3:4 and the area is 150 cm². Find the diagonals.
Solution:
Given:
- d₁ : d₂ = 3 : 4. Let d₁ = 3k and d₂ = 4k.
Using the area formula:
- 150 = (1/2) x 3k x 4k
- 150 = (1/2) x 12k²
- 150 = 6k²
- k² = 25
- k = 5
Finding diagonals:
- d₁ = 3 x 5 = 15 cm
- d₂ = 4 x 5 = 20 cm
Verification: Area = (1/2) x 15 x 20 = 150 cm². Correct.
Answer: The diagonals are 15 cm and 20 cm.
Example 10: Example 10: Rhombus-shaped garden
Problem: A garden is in the shape of a rhombus with diagonals 40 m and 30 m. Find the cost of planting grass at Rs 5 per m².
Solution:
Area of the garden:
- = (1/2) x 40 x 30 = 600 m²
Cost of planting grass:
- = 600 x 5 = Rs 3000
Answer: The cost is Rs 3,000.
Real-World Applications
The area of a rhombus formula is used in many real-life situations:
- Floor Tiling: Rhombus-shaped tiles are common in decorative flooring. Calculating the area of each tile helps determine how many tiles are needed.
- Land Measurement: Fields or plots shaped like rhombuses require area calculations for buying, selling, or taxation purposes.
- Kite Making: A kite is essentially a rhombus (or close to it). The area formula helps calculate the amount of paper or cloth needed.
- Diamond Cutting: Gemstones are often cut into rhombus-shaped facets. Jewellers use area calculations for pricing.
- Architecture: Rhombus-shaped windows, wall panels, and decorative elements require area calculations for material estimation.
- Road Signs: Diamond-shaped (rhombus) warning signs need area calculations for manufacturing.
- Farming: Rhombus-shaped plots need area calculations for seed requirements, fertiliser quantities, and irrigation planning.
Key Points to Remember
- Area = (1/2) x d₁ x d₂ (using diagonals) — the primary formula.
- Area = base x height (as a parallelogram) — the alternative formula.
- The diagonals of a rhombus bisect each other at right angles.
- The diagonals divide the rhombus into 4 congruent right triangles.
- side² = (d₁/2)² + (d₂/2)² — relates side and diagonals (Pythagoras).
- To find a missing diagonal: d₂ = 2 x Area / d₁.
- If perimeter and one diagonal are given, find the side first (side = P/4), then the other diagonal using Pythagoras.
- The area of a rhombus is always half the product of its diagonals.
- A rhombus is a special parallelogram, so parallelogram area formulas also apply.
- Units of area are always squared (cm², m², etc.).
Practice Problems
- Find the area of a rhombus with diagonals 22 cm and 16 cm.
- The area of a rhombus is 210 cm² and one diagonal is 14 cm. Find the other diagonal.
- A rhombus has side 17 cm and one diagonal 16 cm. Find the area.
- The diagonals of a rhombus are in the ratio 5:12 and the area is 270 cm². Find the diagonals.
- A rhombus has perimeter 68 cm and one diagonal 30 cm. Find the area.
- Find the cost of painting a rhombus-shaped board with diagonals 50 cm and 36 cm at Rs 2 per cm².
- A rhombus-shaped field has area 1200 m². If one diagonal is 60 m, find the other diagonal.
- The diagonals of a rhombus are 10 cm and 24 cm. Find the side and the perimeter.
Frequently Asked Questions
Q1. What is the formula for the area of a rhombus?
Area = (1/2) x d₁ x d₂, where d₁ and d₂ are the lengths of the two diagonals.
Q2. Why do we use diagonals instead of sides?
The diagonals of a rhombus bisect each other at right angles, creating 4 right triangles whose total area equals (1/2) x d₁ x d₂. This formula is simpler than using base and height.
Q3. Can we use base x height for a rhombus?
Yes. Since a rhombus is a parallelogram, Area = base x height also works. Here, the base is the side of the rhombus and the height is the perpendicular distance between two parallel sides.
Q4. How do you find the diagonal if the area and one diagonal are given?
Use d₂ = 2 x Area / d₁. For example, if Area = 100 cm² and d₁ = 20 cm, then d₂ = 200/20 = 10 cm.
Q5. How do you find the area if the side and one diagonal are given?
Use Pythagoras theorem: side² = (d₁/2)² + (d₂/2)². Solve for d₂, then use Area = (1/2) x d₁ x d₂.
Q6. Are the diagonals of a rhombus always unequal?
The diagonals are unequal in a general rhombus. They are equal only when the rhombus is a square.
Q7. What is the area of a rhombus with diagonals 0?
If either diagonal is 0, the area is 0, and the shape degenerates — it is no longer a proper rhombus.
Q8. How is the area of a rhombus related to its side?
Directly: Area = side x height. Or using diagonals and the Pythagoras relationship: once you know the side and one diagonal, you can find the other diagonal and then the area.
Q9. Is the area of a rhombus always less than the area of a square with the same side?
Yes. For the same side length, the square (which has the maximum possible area for any rhombus) has a larger area than a non-square rhombus. This is because the height of a non-square rhombus is less than its side.
Q10. What units are used for the area of a rhombus?
Square units — cm², m², km², etc. If the diagonals are in cm, the area is in cm².
