What is the formula of area of rhombus?

A rhombus is a special quadrilateral where all four sides are equal in length. It's also known as a diamond shape. Key properties include:

• All sides are equal
• Opposite angles are equal
• Diagonals bisect each other at right angles
• Diagonals divide the rhombus into four congruent right triangles

Complete Formula Reference Table

Detailed Formula Explanations

1. Area Using Diagonals (Primary Formula)

Formula:Area = ½ × d₁ × d₂

Explanation: This is the most commonly used formula for finding the area of a rhombus. Since the diagonals of a rhombus bisect each other at right angles, they divide the rhombus into four congruent right triangles.

Example: If d₁ = 12 cm and d₂ = 8 cm

• Area = ½ × 12 × 8 = 48 cm²

2. Area Using Base and Height

Formula:Area = base × height

Explanation: Like any parallelogram, the area equals base times height. The height is the perpendicular distance between two parallel sides.

Example: If side = 10 cm and height = 6 cm

• Area = 10 × 6 = 60 cm²

3. Area Using Side and Angle

Formula:Area = a² × sin(θ)

Explanation: When you know the side length and any interior angle, use this formula. Since opposite angles are equal in a rhombus, you can use any known angle.

Example: If side = 8 cm and angle = 60°

• Area = 8² × sin(60°) = 64 × (√3/2) = 32√3 cm²

4. Area Using Side and One Diagonal

Formula:Area = a × √(4a² - d²)/2

Explanation: This formula is derived using the Pythagorean theorem. If you know one side and one diagonal, you can find the other diagonal using the relationship between them.

Step-by-step derivation:

1. In a rhombus, diagonals bisect at right angles
2. Each quarter forms a right triangle with side 'a' as hypotenuse
3. If one diagonal is 'd₁', half of it is d₁/2
4. Using Pythagorean theorem: (d₂/2)² + (d₁/2)² = a²
5. Solve for the other diagonal and apply the diagonal formula

Grade-Specific Applications

Class 8 Level

Focus: Basic diagonal formula and base-height method

• Primary Formula: Area = ½ × d₁ × d₂
• Alternative: Area = base × height
• Practice: Simple numerical problems with given measurements

Class 9-10 Level

Additional Concepts:

• Introduction to trigonometric relationships
• New Formula: Area = a² × sin(θ)
• Applications: Real-world problems involving rhombus shapes

Class 11-12 Level

Advanced Applications:

• Complex Formula: Area = a × √(4a² - d²)/2
• Coordinate geometry: Finding area using coordinate methods
• Integration: Combining multiple geometric concepts

Related Formulas and Properties

Perimeter of Rhombus

Formula:Perimeter = 4a

• Where 'a' is the length of one side

Relationship Between Diagonals and Side

Formula:d₁² + d₂² = 4a²

• This helps find missing measurements

Height of Rhombus

Formula:height = Area/side = (d₁ × d₂)/(2a)