# Rhombus: Definition, Properties, Formula, 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 learn about rhombus, its properties, examples, formula, etc. in detail.

## 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 actually 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.

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.

## 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.

Check Out, Rhombus Is Not A Square

## Rhombus Properties

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 Ã— d2) Â 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 Ã— 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 = 4Ã—s

where s is the side of Rhombus

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

Read More On, 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=d1Ã—d2, where d 1 and d2 are the lengths of the diagonals.

Read More On Why diagonals of rhombus are not equal

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

Features Rhombus Square Rectangle Parallelogram Trapezoid Difference between Rhombus and Other Quadrilaterals 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 90Â° All angles are 90Â° 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 Ã— Height or 1/2Ã—ProductÂ ofÂ diagonals SideÂ² Length Ã— Width Base Ã— Height 12Ã—(SumÂ ofÂ parallelÂ sides)Ã—Height21â€‹Ã—(SumÂ ofÂ parallelÂ sides)Ã—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

## 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?Â

Solution:

Perimeter = 40 cm

Perimeter = 4 Ã— side

40 = 4Ã—side

â‡’ side(base) = 10 cm

and height = 12 cm (given)

Now, Area of Rhombus = base Ã— height

â‡’ Area = 10Ã—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 =Â \frac{(2x+2)(4x+4)}{2}Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Â

â‡’ A =Â \frac{\sqrt{8x^2}}{2}

â‡’ A =Â \frac{8x^2+16x+8}{2}

â‡’ A = (4x2 + 8x + 4) unit2

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

Solution:

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

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

A =Â \frac{\sqrt{2x}\sqrt{4x}}{2}

â‡’ A =Â x\sqrt{2}Â Â Â Â Â Â Â  Â cm2Â

## Rhombus Practice Exercise

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

### 1. 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.

### 2. 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.

### 3. Are all 4 sides of a rhombus equal?

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

### 4. 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.

### 5. 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

### 6. 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.

### 7. Are rhombus diagonals equal?

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

### 8. What shape has 4 equal sides and equal-length diagonals?

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

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