Rhombus: Detailed Explanation
A rhombus is a special type of quadrilateral (four-sided figure) that is also part of the parallelogram family. It is often referred to as a “diamond” shape. Below are the key properties and details about the rhombus:
Properties of a Rhombus:
- All sides are equal: One of the most defining characteristics of a rhombus is that all four sides are equal in length.
- Opposite angles are equal: While the angles are not necessarily 90°, the opposite angles of a rhombus are always equal.
- Opposite sides are parallel: Like all parallelograms, a rhombus has opposite sides that are parallel.
- Diagonals:
- The diagonals of a rhombus bisect each other at right angles (90°).
- The diagonals do not have to be equal in length.
- The diagonals bisect the internal angles of the rhombus.
- Area and Perimeter:
- Area = 12×Diagonal1×Diagonal2\frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2 (The area is calculated using the lengths of the diagonals).
- Perimeter = 4 × Side (If the side of a rhombus is ‘a,’ the perimeter is 4a4a).
- Symmetry:
- A rhombus has two lines of symmetry: one along each diagonal.
- A rhombus also has rotational symmetry at 180°.
Comparison of Rhombus with Other Figures:
| Figure | Similarities with Rhombus | Differences from Rhombus |
|---|---|---|
| Square | – All sides are equal. – Opposite sides are parallel. – Diagonals bisect each other at 90°. | – A square has all angles equal to 90°, while a rhombus does not necessarily have 90° angles. – Diagonals of a square are equal, whereas in a rhombus they are not. |
| Rectangle | – Opposite sides are parallel. | – In a rectangle, only opposite sides are equal, while in a rhombus all sides are equal. – The diagonals of a rectangle are equal, but they do not bisect at right angles. – All angles in a rectangle are 90°, unlike in a rhombus. |
| Parallelogram | – Opposite sides are parallel and equal. – Opposite angles are equal. | – In a parallelogram, opposite sides are equal but not necessarily all four sides. – Diagonals do not bisect each other at right angles in a parallelogram. |
| Trapezium (Trapezoid) | – Both are quadrilaterals. | – In a trapezium, only one pair of sides is parallel, while in a rhombus both pairs of sides are parallel. – Sides and angles in a trapezium are not necessarily equal. |
Rhombus vs Square:
- Commonality: Both the rhombus and square have all sides equal, opposite sides parallel, and diagonals that bisect each other at 90°.
- Difference: The angles in a square are all 90°, while in a rhombus, the angles can vary. Also, the diagonals of a square are equal, while in a rhombus, they are not.
Rhombus vs Rectangle:
- Commonality: Both have opposite sides that are parallel, and both are types of parallelograms.
- Difference: In a rectangle, opposite sides are equal, but not all sides are equal like in a rhombus. The angles in a rectangle are all 90°, whereas in a rhombus, the angles can vary.
Rhombus vs Parallelogram:
- Commonality: Both have opposite sides parallel and equal, and opposite angles are equal.
- Difference: In a parallelogram, only opposite sides are equal, while in a rhombus, all four sides are equal. The diagonals of a parallelogram do not bisect each other at right angles, whereas in a rhombus, they do.
Rhombus vs Trapezium:
- Commonality: Both are quadrilaterals (four-sided figures).
- Difference: A trapezium has only one pair of parallel sides, while a rhombus has two pairs. In a trapezium, sides and angles are not necessarily equal, whereas in a rhombus, all sides are equal.
Real-Life Examples of a Rhombus:
- Diamond shape: The shape of diamonds often seen in playing cards or patterns.
- Kites: Some kites are designed in the shape of a rhombus.
- Windows: Certain types of window panes are rhombus-shaped.
Important Formulas:
- Area = 12×Diagonal1×Diagonal2\frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2.
- Perimeter = 4 × Side.
- Diagonal Lengths: The diagonals are unequal, but they intersect at right angles.