Rhombus: Definition, Properties, Formula and Examples

Rhombus is a quadrilateral with all four sides equal and opposite sides parallel to each other. The opposite angles of a rhombus are equal. Any rhombus can be considered a parallelogram, but not all parallelograms are rhombus.

Let’s know more about Rhombus and it’s properties, examples and formula in detail below.

Rhombus

A rhombus is a special case of a quadrilateral known as a parallelogram. where the adjacent sides are equal in length and also the diagonals bisect each other at right angles. We can also state that a rhombus is a square when all its angles are equal to 90 degrees.

The plural form of a rhombus is rhombi or rhombuses.

Rhombus Definition

A rhombus is a quadrilateral with all sides of equal length and opposite sides parallel, but typically with unequal angles.

Rhombus Shape

The diagonals of a rhombus bisect each other at right angles. That is, they intersect at a 90-degree angle and divide each other into two equal segments. Moreover, the diagonals of a rhombus are perpendicular bisectors of each other, meaning they divide each other into equal parts and form right angles at their intersection point. The diagonals of Rhombus are not necessarily equal in length. However, they bisect each other at their midpoint, creating four right triangles with equal hypotenuses (the sides of the rhombus).

Rhombus Symmetry: A rhombus exhibits symmetry across its diagonals. This means that if you fold a rhombus along one of its diagonals, the two resulting halves will perfectly overlap each other.

The figure below shows a rhombus shape where AB = BC = CD = DA and the diagonals AC and BD bisect each other at a right angle. This confirms its classification as a quadrilateral.

Read More

• Parallelograms

Rhombus Examples

Rhombus is a very common shape and can be seen in a variety of objects which we use in our daily lives. Various Rhombus-shaped objects are Jewelry, Kites, Sweets, Furniture, etc.

Note: All squares are rhombuses, but not all rhombuses are squares. This is because a square is a special type of rhombus that has all four sides equal in length and all four angles equal to 90 degrees. However, a rhombus can have angles that are not equal to 90 degrees.

Is Square a Rhombus?

Yes, a square is a special type of rhombus. By definition, a rhombus is a quadrilateral with all four sides of equal length. A square fits this definition perfectly because it has four equal sides.

Also Read

• Rhombus Is Not A Square

Properties of Rhombus

The properties of a rhombus are:

• All the sides of a rhombus are equal. In fact, it is just a parallelogram with equal adjacent sides.
• All Rhombus has two diagonals, which connect the pairs of opposite vertices. A rhombus is symmetrical along both its diagonals. The diagonals of a rhombus are perpendicular bisectors to each other.
• In the event of all the angles of a rhombus are equal, it is called a square.
• The diagonals of a rhombus would always bisect each other at a 90 degrees angle.
• Not only do the diagonals bisect each other, but they also bisect the angles of a rhombus.
• The two diagonals of a rhombus divide it into four right-angled congruent triangles.
• There cannot be a circumscribing circle around a Rhombus.
• It is impossible to have an inscribing circle inside a rhombus.

Rhombus Formula

A rhombus is characterized by its equal-length sides and interesting geometric properties. The formulas associated with a rhombus are important for various mathematical calculations.

These are some important formulas related to Rhombus :

• Area
• Perimeter

Area of Rhombus

The area of the Rhombus is the space enclosed by all four boundaries of the Rhombus it is measured in unit squares. There are two ways of finding Areas of a Rhombus which are discussed below

1.) Area of Rhombus when both Diagonals are given

The area of the rhombus is the region covered by it in a two-dimensional plane. The formula for the area is equal to the product of the diagonals of the rhombus divided by 2. It can be represented as:

Area of Rhombus = 1/2(d₁ × d₂)  Sq. unit

where d1 and d2 are diagonals of a rhombus.

2.) Area of Rhombus when Base and Altitude are given

When the Base and Altitude of a Rhombus are given then the formula calculates its area:

Area of Rhombus = Base × Height

Perimeter of Rhombus

Perimeter of a rhombus is defined as the sum of all its sides. Since all the sides of a rhombus are equal in length, it can be said that the Perimeter of a Rhombus is four times the length of one side.

Thus, if s denotes the length of a side of a rhombus,

Perimeter of Rhombus = 4×s

where s is the side of Rhombus

For instance, if each side of a rhombus measures 5 cm, its perimeter would be 4×5 cm, equating to 20 cm.

Read More

• Formulas for Rhombus

Diagonals of a Rhombus

The diagonals of a rhombus bisect each other at right angles. It means that they intersect at a 90-degree angle, a property not shared by all quadrilaterals.

• This perpendicular intersection results in the diagonals dividing the rhombus into four congruent right-angled triangles.
• While the sides of a rhombus are of equal length, its diagonals are generally of different lengths and they bisect the internal angles of the rhombus.
• Each diagonal cuts an angle of the rhombus into two equal parts.
• The lengths of the diagonals can be used to calculate the area of the rhombus, with the formula

Area=d₁×d₂, where d ₁ and d₂ are the lengths of the diagonals.

Read More

• Why diagonals of Rhombus are not equal

Rhombus vs Other Quadrilaterals

Let’s see the comparison of rhombus with other common quadrilaterals in the table below.

Difference between Rhombus and Other Quadrilaterals
Features	Rhombus	Square	Rectangle	Parallelogram	Trapezoid
Sides	All sides equal length	All sides equal length	Opposite sides equal	Opposite sides equal	Only one pair of opposite sides parallel
Angles	Opposite angles equal	All angles are 90°	All angles are 90°	Opposite angles equal	No specific angle properties
Diagonals	Bisect each other at right angles and are not equal	Bisect each other at right angles and are equal	Bisect each other but not at right angles and are equal	Bisect each other but not at right angles and are not equal	No specific diagonal properties
Symmetry	Both line and rotational symmetry	Both line and rotational symmetry	Line symmetry	Line symmetry	Typically no line or rotational symmetry
Parallel Sides	Opposite sides are parallel	All sides are parallel	Opposite sides are parallel	Opposite sides are parallel	Only one pair of opposite sides parallel
Area Formula	Base × Height or 1/2×Product of  diagonals	Side²	Length × Width	Base × Height	12×(Sum of parallel sides)×Height21​×(Sum of parallel sides)×Height
Special Properties	All sides are equal and it is a parallelogram	All properties of a rectangle and a rhombus	Diagonals are equal and bisect each other	Opposite sides are equal and parallel, opposite angles are equal	Only one pair of opposite sides is required to be parallel

Also Read

• Difference between Rhombus Diamond and Trapezoid

Rhombus Example Questions

Let’s solve some example questions on Rhombus and its properties.

Example 1: MNOP is a rhombus. If diagonal MO = 29 cm and diagonal NP = 14 cm, What is the area of rhombus MNOP?

Solution:

Area of a rhombus = (d₁)(d₂)/2

Substituting the lengths of diagonals in the above formula, we have:

A = (29)(14)/2 = 406/2 = 203 cm²

Area of rhombus MNOP = 203 cm²

Example 2: ABCD is a rhombus. The perimeter of ABCD is 40, and the height of the rhombus is 12. What is the area of ABCD? 

Solution:

Perimeter = 40 cm

Perimeter = 4 × side

40 = 4×side

⇒ side(base) = 10 cm

and height = 12 cm (given)

Now, Area of Rhombus = base × height

⇒ Area = 10×12 = 120 cm²

Thus, Area of rhombus ABCD is equal to 120 cm²

Example 3: Find the area of a rhombus with diagonal lengths of (2x+2) and (4x+4) units.

Solution:

We know, Area of a rhombus = (d₁)(d₂)/2

Substituting the lengths of diagonals in the above formula, we have:

A = \frac{(2x+2)(4x+4)}{2}                              

⇒ A = \frac{\sqrt{8x^2}}{2}

⇒ A = \frac{8x^2+16x+8}{2}

⇒ A = (4x² + 8x + 4) unit²

Example 4: Find the area of a rhombus if its diagonal lengths are \sqrt{2x}                    cm and \sqrt{4x}                     cm.

Solution:

We know, Area of a rhombus = (d₁)(d₂)/2

Substituting the lengths of diagonals in the above formula, we have:

A = \frac{\sqrt{2x}\sqrt{4x}}{2}

⇒ A = x\sqrt{2}         cm² 

Rhombus Practice Questions

Here are some exercise questions on rhombus for you to solve:

1. If one angle of a rhombus is 60 degrees, what are the measures of the other three angles?

2. The diagonals of a rhombus are 10 cm and 24 cm long. Calculate the area of the rhombus.

3. In a rhombus, each diagonal measures 16 cm, and they intersect at right angles. Find the length of each side of the rhombus.

4. A rhombus-shaped garden has a side length of 15 meters and one of its diagonals is 20 meters long. Calculate the area of the garden.

5. In a rhombus, the diagonals intersect at a point that divides each diagonal into segments of 5 cm and 15 cm. Find the lengths of the diagonals.

Rhombus – FAQs

What is Rhombus in Geometry?

A rhombus is a 2-D shape with four sides hence termed as a quadrilateral. It has two diagonals that bisect each other at right angles.

What shape is a rhombus?

A rhombus has a flat two-dimensional shape. It is a type of quadrilateral shape with four sides of equal length.

Are all 4 sides of a rhombus equal?

Yes, all four sides of a rhombus are equal in length.

What are the 4 properties of a rhombus?

The four properties of a rhombus are:

• all four sides are of equal length,
• opposite angles are equal in measure,
• diagonals bisect each other at right angles, and
• consecutive angles are supplementary.

Is a rhombus a square?

A rhombus becomes a square only when all four angles are equal to 90 degrees. Every square is a rhombus but all rhombuses are not Squares

What are the 8 properties of a rhombus?

The eight properties of a rhombus are:

• all four sides are of equal length,
• opposite angles are equal in measure,
• diagonals bisect each other at right angles,
• consecutive angles are supplementary,
• diagonals are of equal length,
• the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals,
• the area is equal to half the product of the diagonals, and
• the perimeter is equal to four times the length of one side.

Are rhombus diagonals equal?

Yes, the diagonals of a rhombus are of equal length.

What shape has 4 equal sides and equal-length diagonals?

A shape with 4 equal sides and equal length diagonals is a square.