How to Calculate the Area of a Rhombus

Area of a Rhombus Using Diagonals: Formula & Steps

Understanding the area of a rhombus is an essential part of geometry for students from middle to senior classes. This concept not only helps in school and board exams but is also important for competitive exams like JEE and Olympiads. Mastery of this topic enables students to solve a variety of questions involving quadrilaterals and related shapes.

What is a Rhombus?

A rhombus is a special type of quadrilateral in which all four sides are equal in length, and opposite sides are parallel. Its diagonals bisect each other at right angles (90°), dividing the rhombus into four right-angled triangles. While every square is a rhombus, not every rhombus is a square, since a rhombus may have unequal angles.

Four equal sides
Opposite angles are equal
Diagonals bisect each other at 90°
Diagonals are not necessarily equal

Recognizing these properties is important while applying formulas or solving geometry problems. At Vedantu, we simplify concepts like the area of a rhombus through visuals and stepwise explanations.

Area of a Rhombus: Formulae

There are two main formulas for finding the area of a rhombus, depending on what information you have:

Using Diagonals: If you know the lengths of the diagonals (d₁ and d₂),
Area = (1/2) × d₁ × d₂
Using Side and Height: If you know the length of a side (a) and the perpendicular height (h),
Area = a × h

Formula	Description	Variables
Area = (1/2) × d₁ × d₂	When diagonals are known	d₁ = diagonal 1 d₂ = diagonal 2
Area = side × height	When side and height are known	side = length of one side height = perpendicular distance from a side to its opposite

Step-by-Step Example Problems

Example 1: Area using Diagonals

Find the area of a rhombus whose diagonals are 10 cm and 8 cm.

Write the formula: Area = (1/2) × d₁ × d₂
Substitute values: Area = (1/2) × 10 × 8 = (1/2) × 80 = 40 cm²

Example 2: Area using Side and Height

A rhombus has a side of 12 cm and a height of 7 cm. Find its area.

Formula: Area = side × height
Area = 12 × 7 = 84 cm²

Example 3: Area when Only Side and Diagonal Are Known

A rhombus has each side of length 13 cm. One of its diagonals is 24 cm. Find the area.

Diagonals bisect each other at right angles. Let diagonal d₁ = 24 cm. So, each half = 12 cm.
Each half-diagonal with side forms a right triangle. Use Pythagoras’ theorem:
side² = (half d₁)² + (half d₂)²
13² = 12² + x² ⇒ 169 = 144 + x²
x² = 25 ⇒ x = 5
The other diagonal, d₂ = 2 × 5 = 10 cm.
Area = (1/2) × 24 × 10 = 120 cm²

Practice Problems

Find the area of a rhombus whose diagonals are 15 cm and 18 cm.
A rhombus has a side of 10 cm and a height of 8 cm. What is its area?
If the area of a rhombus is 60 cm² and one diagonal is 12 cm, find the length of the other diagonal.
Each side of a rhombus is 5 cm, and one angle is 60°. Find the area. (Recall: Area = side² × sin(angle))
The diagonals of a rhombus are in the ratio 3:4 and the area is 216 cm². Find the length of each diagonal.

Common Mistakes to Avoid

Mixing up the formulas for a rhombus and a square (area of a square is side × side).
Forgetting to halve the product of diagonals when using the diagonal formula.
Using the wrong measurement for the height (must be perpendicular from the base).
Not converting units properly (example: mixing cm and m).

Real-World Applications

The area of a rhombus is used in finding the area of kites, in construction tiling or flooring patterns, land measurements, and in some engineering designs involving symmetrical structures. Understanding this formula is valuable in design, architecture, and practical geometry problems.

Related Concept Links

In this topic, we explored the area of a rhombus using formulas based on diagonals and side-height, discussed worked examples, common errors, and real-world uses. Mastering these concepts will help you tackle geometry problems efficiently. For more concepts and easy explanations, explore related topics with Vedantu.