Rhombus in Maths: Definition, Properties & Formulas

The concept of rhombus plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding the rhombus will help you easily solve geometry, mensuration, and logical reasoning questions in school and competitive exams.

What Is Rhombus?

A rhombus is defined as a special type of quadrilateral where all four sides are equal in length and opposite sides are parallel. You’ll find this concept applied in areas such as geometry (especially quadrilaterals), mensuration, and even symmetry in real-life objects like tiles and kites.

Definition and Properties of Rhombus

In simple terms, a rhombus is a flat, four-sided shape with equal-length sides. Opposite sides are parallel, and opposite angles are equal. Importantly, the diagonals of a rhombus bisect each other at right angles (90°) but are generally not equal in length. The rhombus is also known as a “diamond” shape, like the one seen in playing cards.

Key Properties of Rhombus

• All sides are equal in length
• Opposite sides are parallel
• Opposite angles are equal
• Diagonals bisect each other at right angles (90°)
• Diagonals bisect opposite angles
• Sum of any two adjacent angles is 180° (supplementary)
• The sum of all interior angles is 360°
• The rhombus has 2 lines of symmetry along its diagonals

Rhombus vs. Parallelogram, Square, and Kite

Students often get confused between rhombus, square, and parallelogram. While all squares and rhombuses are parallelograms, not all parallelograms are rhombuses or squares. A square is a special rhombus where all angles are 90°. A kite has two distinct pairs of adjacent sides that are equal, but in a rhombus, all four sides are equal.

Key Formula for Rhombus

Here are the standard formulas:

• Area: \( \text{Area} = \frac{d_1 \times d_2}{2} \), where \(d_1\) and \(d_2\) are the lengths of the diagonals.
• Perimeter: \( \text{Perimeter} = 4a \), where \(a\) is the length of one side.

You can also calculate area using base and height: \( \text{Area} = \text{base} \times \text{height} \).

Diagonals, Angles, and Symmetry of Rhombus

• The diagonals of a rhombus intersect at 90° and bisect each other.
• Each diagonal cuts the rhombus into two congruent triangles.
• Opposite angles are equal; the adjacent angles add up to 180°.
• A rhombus has exactly 2 lines of symmetry (its diagonals) and rotational symmetry of order 2 (180°).

Step-by-Step Illustration: Finding the Area

1. Suppose the lengths of diagonals are: \( d_1 = 10 \) cm, \( d_2 = 8 \) cm.
2. Use the area formula: \( \text{Area} = \frac{d_1 \times d_2}{2} \).
3. Substitute the values: \( \text{Area} = \frac{10 \times 8}{2} = \frac{80}{2} \).
4. Final answer: Area = 40 cm²

Speed Trick or Vedic Shortcut

To quickly find the area of a rhombus when only the side and one diagonal are given, recall the Pythagoras theorem. If side = a and diagonal = d₁, then:

1. Let halves of the diagonals be \( \frac{d_1}{2} \) and \( x \).
2. By Pythagoras: \( a^2 = (\frac{d_1}{2})^2 + x^2 \)
3. Solve for x, then full second diagonal \( d_2 = 2x \), and use the area formula.

This shortcut helps with tricky questions and is often included in quick revision lists for exams.

Try These Yourself

• Find the area of a rhombus with diagonals 6 cm and 12 cm.
• If the side of a rhombus is 13 cm and one diagonal is 24 cm, find the other diagonal.
• List all the ways a rhombus is different from a square.
• Draw a rhombus and show its lines of symmetry.

Frequent Errors and Misunderstandings

• Thinking diagonals are always equal (true only for squares).
• Confusing rhombus angles as always 90° — only true for squares.
• Mixing up the properties of rhombus with parallelogram or kite.

Relation to Other Concepts

The idea of rhombus connects closely with topics such as quadrilaterals, squares and rectangles, and parallelograms. Mastering this helps with understanding symmetry, area, and coordinate geometry in later chapters.

Classroom Tip

A quick way to remember rhombus is the “all sides equal, but angles not always right” rule. Draw the diamond symbol from cards to associate the look with the properties. Vedantu’s teachers often use this visual approach in live classes for faster retention.

We explored rhombus—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.

Useful Internal Links

• Difference Between Rhombus and Parallelogram
• Quadrilaterals
• Symmetry in Maths
• Area and Perimeter