What is a Rhombus – Definition, Types, Properties & Examples

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The term ‘rhombus’ has been derived from the ancient Greek word ‘rhombos‘, which actually means something that spins. A rhombus is a closed 2D plane figure. It’s also called an equilateral quadrilateral since all of its sides are equal in length.

Let’s understand what is a rhombus and what are the properties of rhombus with examples.

What is a Rhombus?

A rhombus is a special case of a parallelogram. In a rhombus, opposite sides are parallel and the opposite angles are equal and thus satisfy the conditions of a parallelogram. Moreover, all the sides of a rhombus are equal in length, and the diagonals bisect each other at right angles. The shape of a rhombus is a diamond shape. Hence, it is also called a diamond. The plural form of a rhombus is rhombi or rhombuses.

In the above figure, $\text{ABCD}$ is a rhombus, where the sides $\text{AB}$, $\text{BC}$, $\text{CD}$ and $\text{DA}$ are equal and $\text{AC}$ and $\text{D}$ are the its diagonals.

Is Square a Rhombus?

A square can be considered as a special case of a rhombus because it has four equal sides and all the angles of a square are right angles. But the angles of a rhombus need not necessarily have to be right angles. And, hence a rhombus with right angles can be considered a square. Hence, we can say that all squares are rhombuses, but all rhombi or rhombuses are not squares.

In the above figure, 

• $\text{ABCD}$ is a rhombus, where $\text{AB} = \text{BC} = \text{CD} = \text{DA}$, and $\angle \text{A}$, $\angle \text{B}$, $\angle \text{C}$, and $\angle \text{D}$ are not $90^{\circ}$. (Angles not necessarily $90^{\circ}$).
• $\text{PQRS}$ is a square, where $\text{PQ} = \text{QR} = \text{RS} = \text{SP}$, and $\angle \text{P} = \angle \text{Q}= \angle \text{R} = \angle \text{S} = 90^{\circ}$. (Angles are necessarily of $90^{\circ}$).

Properties of a Rhombus

A rhombus is considered to be one of the special parallelograms as it has all the properties of a parallelogram. A rhombus has two diagonals as its two lines of symmetry. The following are the characteristic properties of a parallelogram.

• The rhombus has four interior angles.
• The sum of the interior angles of a rhombus is $360^{\circ}$.
• Opposite angles are congruent or equal.
• Adjacent angles are supplementary.
• The opposite sides are equal and parallel.
• In a rhombus, diagonals bisect each other at right angles.
• The diagonals of a rhombus bisect the interior angles.

The Diagonals of a Rhombus Bisect the Interior Angles

Let’s consider a rhombus $\text{ABCD}$, where $\text{AC}$ is one of its diagonals.

The diagonal $\text{AC}$ bisects $\angle \text{A}$ and $\angle \text{C}$. 

Let’s prove the above statement.

In $\triangle \text{ABC}$ and $\triangle \text{ADC}$ 

$\text{AC} = \text{AC}$ (Common)

$\text{AB} = \text{CD}$ (Sides of a rhombus)

$\text{BC} = \text{AD}$ (Sides of a rhombus)

Therefore, $\triangle \text{ABC} \cong \triangle \text{ADC}$  (SSS congruency criterion)

Thus, $\angle {BAC} = \angle {DAC}$ and $\angle {BCA} = \angle {DCA}$ (Corresponding Parts of Congruent Triangles) 

Hence $\text{AC}$ bisects $\text {A}$ and $\angle {C}$.

Similarly, by joining $\text{BD}$, we can prove that $\text{BD}$ bisects $\text {B}$ and $\angle {D}$.

Figure Formed by Joining the Midpoint of the Sides of a Rhombus is a Rectangle

Let’s consider a rhombus $\text{ABCD}$ with $\text{P}$, $\text{Q}$, $\text{R}$ and $\text{S}$ as the midpoints of $\text{AB}$, $\text{BC}$, $\text{CD}$ and $\text{DA}$.

The figure $\text{PQRS}$ is a rectangle. 

Let’s prove the above statement.

From the property of a rhombus, $\text{AC} || \text{PQ}$

In $\triangle \text{ACB}$ and $\triangle \text{PQB}$

$\angle \text{BAC} = \angle \text{BPQ}$ and $\angle \text{BCA} = \angle \text{BQP}$ (Corresponding angles are equal)

$\angle \text{ABC} = \angle \text{PBQ}$ (the same angles)

Therefore, $\triangle \text{ABC} \cong \triangle \text{ PBQ}$

$=> \frac{ \text{AC}}{\text{PQ}} = \frac{\text{BC}}{\text{BQ}}$

$=>\text{PQ} = \frac{1}{2} \text{AC}$

Now consider $\triangle \text{BCD}$ and $\triangle \text{RCQ}$

$\text{RQ} = \frac{1}{2} \text{BD}$

Similarly in $\triangle \text{ADC}$ and $\triangle \text{SDR}$

$\text{SR} = \frac{1}{2} \text{AC}$

$\text{SP} = \frac{1}{2} \text{BD}$

Since the opposite sides are equal, therefore $\text{PQ} = \text{SR}$ and $\text{RQ} = \text{SP}$

We know that

$\angle \text{APS} + \angle \text{SPQ} + \angle \text{BPQ} = 180^{\circ}$

$\angle \text{ABD} + \angle \text{SPQ} + \angle \text{BAC} = 180^{\circ}$

$\frac{1}{2} \angle \text{ABC} + \angle \text{SPQ} + \frac{1}{2} \angle \text{BAD} = 180^{\circ}$

$\angle \text{SPQ} + \frac{1}{2}  \left( \angle \text{ABC} + \angle \text{BAD} \right) = 180^{\circ}$

Using the property of rhombus, we get

$\angle \text{SPQ} + \frac{1}{2} \times 180^{\circ} = 180^{\circ}$

$=> \angle \text{SPQ} = 90^{\circ}$

$=> \angle \text{PQR} = \angle \text{QRS} = \angle \text{RSP} = 90^{\circ}$

Opposite sides are equal and all angles are $90^{\circ}$

So $\text{PQRS}$ is a rectangle.

Facts About Rhombus

The following are some more facts about rhombus.

• You get a cylindrical surface having a convex cone at one end and a concave cone at another end when the rhombus is revolved about any side as the axis of rotation.
• You get a cylindrical surface having concave cones on both ends when the rhombus is revolved about the line joining the midpoints of the opposite sides as the axis of rotation.
• You get solid with two cones attached to their bases when the rhombus is revolving about the longer diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the shorter diagonal of the rhombus.
• You get solid with two cones attached to their bases when the rhombus is revolving about the shorter diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the longer diagonal of the rhombus.

Practice Problems

1. What is a rhombus?
2. Under what condition a parallelogram becomes a rhombus?
3. Under what condition a rhombus becomes a square?
4. State True or False
   • The diagonals of a rhombus are equal.
   • The diagonals of a rhombus bisect each other.
   • The diagonals of a rhombus intersect at $90^{\circ}$.
   • The diagonals of a rhombus bisect the internal angles.
   • The figure formed by joining the midpoints of the sides of a rhombus is a square.
   • The figure formed by joining the midpoints of the sides of a rhombus is a rhombus.
   •  The figure formed by joining the midpoints of the sides of a rhombus is a rectangle.

FAQs

Conclusion

A rhombus is a special type of parallelogram in which opposite sides are parallel, and the opposite angles are equal. The diagonals of a rhombus bisect each at right angles. An important point to remember is that all squares are rhombi but all rhombi are not squares.

Recommended Reading

• How to Construct a Triangle(With Steps, Diagrams & Examples)
• Median of a Triangle(Definition & Properties)
• Altitude of a Triangle(Definition & Properties)
• Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
• Similarity of Triangles Criteria – SSS, SAS, AA
• Angle Bisector of a Triangle – Definition, Properties & Examples
• What is Quadrilateral in Math(Definition, Shape & Examples)
• Properties of Triangle – Theorems & Examples
• How to Construct a Triangle(With Steps, Diagrams & Examples)
• Median of a Triangle(Definition & Properties)
• Altitude of a Triangle(Definition & Properties)
• Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
• Similarity of Triangles Criteria – SSS, SAS, AA
• Types of Triangles – Definition & Examples
• What is Triangle in Geometry – Definition, Shapes & Examples
• Pair of Angles – Definition, Diagrams, Types, and Examples
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• What is an Angle in Geometry – Definition, Properties & Measurement
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