1. Definition of a Rhombus
A rhombus is a quadrilateral in which all four sides are equal. It looks like a ‘diamond’ shape, but mathematically it is also a special type of parallelogram because its opposite sides are parallel.
If a rhombus is named as \(ABCD\), then:
\(AB = BC = CD = DA\)
2. Properties of a Rhombus
A rhombus has many unique properties involving angles and diagonals. These help in geometric proofs and numerical problems.
2.1. All Sides are Equal
Every side of a rhombus has the same length. This is the key property that differentiates it from a parallelogram.
2.2. Opposite Angles are Equal
Opposite angles in a rhombus are always equal:
\(\angle A = \angle C\)
\(\angle B = \angle D\)
2.3. Adjacent Angles are Supplementary
Any two angles next to each other add up to:
\(180^\circ\)
2.4. Diagonals are Perpendicular
The diagonals of a rhombus meet at right angles:
\(AC \perp BD\)
2.5. Diagonals Bisect Each Other
The diagonals cut each other into equal halves. They also divide the rhombus into four congruent right triangles.
2.6. Diagonals Bisect Angles
Each diagonal splits the angle at its vertex into two equal angles. This is a property not shared by rectangles.
3. Diagonals of a Rhombus
The diagonals of a rhombus are extremely important because they determine both the shape and the area.
3.1. Diagonal Properties
- Diagonals are perpendicular.
- They bisect each other.
- They bisect the angles of the rhombus.
3.2. Example
If diagonal \(AC = 16\text{ cm}\) and diagonal \(BD = 12\text{ cm}\), then each half is:
\(AO = OC = 8\text{ cm}\)
\(BO = OD = 6\text{ cm}\)
4. Area of a Rhombus
A rhombus has two useful formulas for area. You can use either depending on which measurements are given.
4.1. Area Using Diagonals
If \(d_1\) and \(d_2\) are the lengths of diagonals, then:
\(\text{Area} = \dfrac{1}{2} d_1 d_2\)
4.2. Area Using Base and Height
If side = \(a\) and the height is \(h\), then:
\(\text{Area} = a \times h\)
4.3. Quick Example
If diagonals are \(10\text{ cm}\) and \(24\text{ cm}\):
\(\text{Area} = \dfrac{1}{2}(10)(24) = 120\text{ cm}^2\)
5. Rhombus in the Quadrilateral Family
A rhombus belongs to the parallelogram family and also shares properties with the square.
5.1. Rhombus as a Parallelogram
It has equal and parallel opposite sides, and diagonals that bisect each other—just like a parallelogram.
5.2. Rhombus vs Square
A square is a rhombus with all angles equal to \(90^\circ\). If a rhombus has right angles, it automatically becomes a square.