Area of a Rhombus Calculator
Calculate the area, side length, and perimeter of a rhombus from its two diagonals. See also Area of Parallelogram Calculator and Area of Rectangle Calculator.
How to Calculate the Area of a Rhombus
A rhombus is a quadrilateral with all four sides equal. Its diagonals bisect each other at right angles. To find the area, multiply the two diagonals and divide by 2. The side length can be found from the diagonals using the Pythagorean theorem: side = √((d₁/2)² + (d₂/2)²), since each half-diagonal forms a right triangle with the side.
Rhombus Area Formula
A = ½ × d₁ × d₂
Side = √((d₁/2)² + (d₂/2)²)
Perimeter = 4 × side
Example
Find the area of a rhombus with diagonals 10 and 8:
A = ½ × d₁ × d₂
A = ½ × 10 × 8
A = 40 square units
Side = √(5² + 4²) = √(25 + 16) = √41 ≈ 6.4031
Perimeter = 4 × 6.4031 ≈ 25.6125
Rhombus Area Reference Table
| d₁ | d₂ | Area | Side | Perimeter |
|---|---|---|---|---|
| 4 | 3 | 6.00 | 2.5000 | 10.0000 |
| 6 | 4 | 12.00 | 3.6056 | 14.4222 |
| 8 | 6 | 24.00 | 5.0000 | 20.0000 |
| 10 | 8 | 40.00 | 6.4031 | 25.6125 |
| 12 | 8 | 48.00 | 7.2111 | 28.8444 |
| 14 | 10 | 70.00 | 8.6023 | 34.4093 |
| 16 | 12 | 96.00 | 10.0000 | 40.0000 |
| 20 | 14 | 140.00 | 12.2066 | 48.8262 |
| 24 | 18 | 216.00 | 15.0000 | 60.0000 |
| 30 | 20 | 300.00 | 18.0278 | 72.1110 |
Real-World Applications of Rhombus Area
Kite Design and Manufacturing
Traditional diamond kites are rhombus-shaped. Kite makers need the area to determine how much fabric or paper to cut, and the diagonal measurements help with the structural frame design.
Diamond-Shaped Road Signs
Warning road signs (construction ahead, school zone) are diamond-shaped rhombuses. Calculating the area is needed to determine reflective material requirements and wind load calculations.
Jewelry and Gemstone Cuts
Many gemstone facets and jewelry settings use rhombus shapes. Jewelers calculate the area to determine carat weight, material costs, and the visual impact of the stone.
Decorative Floor and Wall Patterns
Rhombus tiles create stunning geometric patterns in flooring and wall designs. Architects calculate individual tile area to estimate total tiles needed and project costs.
Solved Examples
Example 1: Building a Diamond Kite
You are building a kite with a vertical diagonal of 90 cm and a horizontal diagonal of 60 cm. How much fabric do you need?
d₁ = 90 cm (vertical), d₂ = 60 cm (horizontal)
A = ½ × d₁ × d₂
A = ½ × 90 × 60
A = 2,700 cm² of fabric (add seam allowance)
Example 2: Diamond Road Sign
A standard diamond warning sign has diagonals of 75 cm and 75 cm. What is the reflective area?
d₁ = 75 cm, d₂ = 75 cm (this is a square rotated 45°)
A = ½ × 75 × 75
A = ½ × 5,625
A = 2,812.5 cm² of reflective surface
Example 3: Rhombus Floor Tile
A decorative rhombus tile has diagonals of 24 cm and 16 cm. How many tiles cover a 4.8 m² area?
Tile area = ½ × 24 × 16 = 192 cm² = 0.0192 m²
Tiles needed = 4.8 ÷ 0.0192
250 tiles needed (add 10% waste = 275 tiles)
Practice Questions
1. A rhombus has diagonals of 14 cm and 10 cm. Find the area.
Answer: A = ½ × 14 × 10 = 70 cm²
2. A diamond-shaped garden bed has diagonals 6 m and 4 m. How much mulch area is needed?
Answer: A = ½ × 6 × 4 = 12 m²
3. A rhombus has area 96 cm² and one diagonal of 12 cm. Find the other diagonal.
Answer: d₂ = 2A/d₁ = 2×96/12 = 16 cm
4. A kite (rhombus) has diagonals 1.2 m and 0.8 m. What is the side length?
Answer: side = √((0.6)² + (0.4)²) = √(0.36 + 0.16) = √0.52 ≈ 0.72 m
5. A rhombus has side 13 cm and one diagonal 24 cm. Find the other diagonal and the area.
Answer: Half-diagonal₁ = 12, half-diagonal₂ = √(13² - 12²) = √(169-144) = 5. So d₂ = 10 cm. A = ½ × 24 × 10 = 120 cm²
Key Takeaways
- Rhombus area formula: A = ½ × d₁ × d₂, where d₁ and d₂ are the two diagonals.
- The diagonals of a rhombus always bisect each other at 90° (right angles).
- Side length from diagonals: side = √((d₁/2)² + (d₂/2)²).
- A square is a special rhombus where d₁ = d₂. Its area can also be written as d²/2.
- Alternative formula using side and angle: A = side² × sin(θ), where θ is any interior angle.
- All four sides of a rhombus are equal. Perimeter = 4 × side length.
Frequently Asked Questions
What is a rhombus?
A rhombus is a quadrilateral with all four sides of equal length. It is a special type of parallelogram. Its diagonals bisect each other at right angles (90°).
Is a square a rhombus?
Yes. A square is a special rhombus where all angles are 90°. In a square, the two diagonals are equal in length. The area formula A = ½d₁d₂ still applies.
How do I find the diagonals if I know the side and an angle?
If you know side s and angle θ: d₁ = 2s × sin(θ/2) and d₂ = 2s × cos(θ/2). Then use A = ½d₁d₂ or equivalently A = s² × sin(θ).
What is the relationship between the diagonals?
The diagonals of a rhombus always bisect each other at right angles. They also bisect the vertex angles. The longer diagonal connects the vertices with the smaller angles.