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# Area of Rhombus – Formulas, Methods & Examples In geometry, a rhombus is a special type of parallelogram in which all four sides are equal. Students often get confused with square and rhombus. There lies a difference between a rhombus and a square as in the case of a square the measure of each angle is $90^{\circ}$, whereas, in the case of a rhombus, it’s not.

The region enclosed by the four equal sides of a rhombus is called its area. Let’s understand the procedure of finding the area of a rhombus.

## Rhombus – A 2D Plane Figure

A rhombus is a special type of parallelogram. In a rhombus, opposite sides are parallel and the opposite angles are equal. Moreover, all the sides of a rhombus are equal in length, and the diagonals bisect each other at right angles. The rhombus is also called a diamond or rhombus diamond.

These properties of the rhombus help in distinguishing the rhombus from other quadrilaterals.

• All four sides are equal
• The opposite sides are parallel
• The opposite angles are equal
• The sum of any two adjacent angles is $180^{\circ}$
• The diagonals bisect each other at right angles
• Each diagonal bisects the vertex angles

## Area of a Rhombus

The area of a rhombus is the region enclosed by the four equal sides (perimeter) of a rhombus in a two-dimensional space. It depicts the total number of unit squares that can fit into it and it is measured in square units like $cm^{2}$, $m^{2}$, $ft^{2}$, $in^{2}$.

There are three methods of finding the area of a rhombus

• using diagonals
• using side and height
• using side and angle

## Area of Rhombus Using Diagonals

Consider a rhombus $ABCD$ with diagonals $BD$ and $AC$ of lengths $d_{1}$ and $d_{2}$ respectively. These two diagonals bisect each other at right angles at $O$ forming four congruent right triangles – $\triangle AOB$, $\triangle BOC$, $\triangle COD$ and $\triangle DOA$.

Area of rhombus = Sum of areas of four triangles = (Area of $\triangle AOB$) + (Area of $\triangle BOC$) + (Area of $\triangle COD$) + (Area of $\triangle DOA$).

Area of $\triangle AOB$ = $\frac {1}{2} \times \frac {d_{1}}{2} \times \frac {d_{2}}{2} = \frac {d_{1}\times d_{2}}{8}$.

Note: Area of triangle is $\frac {1}{2} \times \text {Base} \times \text {Height}$.

Similarly, (Area of $\triangle BOC$) = (Area of $\triangle COD$) = (Area of $\triangle DOA$) = $\frac {d_{1}\times d_{2}}{8}$.

Adding areas of four triangles, we get the area of rhombus ABCD = $\frac {d_{1}\times d_{2}}{8} + \frac {d_{1}\times d_{2}}{8} + \frac {d_{1}\times d_{2}}{8} + \frac {d_{1}\times d_{2}}{8} = 4 \times \frac {d_{1}\times d_{2}}{8} = \frac {d_{1}\times d_{2}}{2}$.

Formula for area of rhombus using diagonals $d_{1}$ and $d_{2}$ is given by $A = \frac {d_{1}\times d_{2}}{2}$.

### Examples

Ex 1: Find the area of a rhombus length of whose diagonals are $5 in$ and $8 in$.

Diagonals of a rhombus $d_{1} = 5 in$ and $d_{2} = 8 in$

Area of rhombus = $\frac {d_{1}\times d_{2}}{2} = \frac {5 \times 8}{2} = 20 in^{2}$.

Ex 2: The length of one diagonal of a rhombus is $10 cm$ and its area is $80 cm^{2}$. Find the length of the other diagonal.

Length of one diagonal of rhombus $d_{1} = 10 cm$

Area of rhombus $A = 80 cm^{2}$

Let the length of the other diagonal be $d_{2}$

Area of rhombus  $A = \frac {d_{1}\times d_{2}}{2} => 80 = \frac {10\times d_{2}}{2} => d_{2} = \frac {2 \times 80}{10} = 16 cm$

Therefore, the length of the second diagonal is $16 cm$.

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## Area of Rhombus Using Side & Height

Consider a rhombus ABCD of the length of each side $a$ and the height of rhombus $h$.

Note: The height of a rhombus is a perpendicular distance between a vertex and the opposite side.

The diagonal $BD$ divides the rhombus $ABCD$ into two equal (congruent) triangles – $\triangle ABD$ and $\triangle BCD$.

Area of rhombus = (Area of $\triangle ABD$) + (Area of $\triangle BCD$).

Area of $\triangle ABD = \frac {1}{2} \times a \times h$ and similarly, $\triangle BCD = \frac {1}{2} \times a \times h$

Area of rhombus = $\frac {1}{2} \times a \times h + \frac {1}{2} \times a \times h = 2 \times \frac {1}{2} \times a \times h = a \times h$.

### Examples

Ex 1: Find the area of a rhombus of side $7 cm$ and perpendicular distance from the opposite vertex $9 cm$.

Length of side of rhombus $a = 7 cm$

Height of rhombus $h = 9 cm$

Area of rhombus = $a \times h = 7 \times 9 = 63 cm^{2}$.

Ex 2: Find the length of the side of a rhombus whose area is $90 mm^{2}$ and the distance of the vertex from the opposite side is $10 mm$.

Area of rhombus $A = 90 mm^{2}$

The distance of the vertex from the opposite side is $h = 10 mm$

Let the length of each side of the rhombus be $a$.

Area of rhombus $A = a \times h => 90 = a \times 10 => a = \frac {90}{10} = 9 mm$.

Therefore, the side of a rhombus of area $90 mm^{2}$ and height $10 mm$ is $9 mm$.

## Area of Rhombus Using Side & Angle

Consider a rhombus ABCD, with the length of each side $a$ and measure one angle $\angle ABC = \theta$.

$AC$ and $BD$ are the diagonals and let $AC = d_{1}$ and $BD = d_{2}$.

Since, the diagonals bisect each other at right angles, therefore, $OA = \frac {d_{1}}{2}$ and $OB = \frac {d_{2}}{2}$

Also, the $\angle ABC$ is bisected by the diagonal $BD$, therefore, $\angle ABD = \angle ABO = \frac {\theta}{2}$

In $\triangle AOB$, $\sin \frac {\theta}{2} = \frac {\frac {d_{1}}{2}}{a}$ and $\cos \frac {\theta}{2} = \frac {\frac {d_{2}}{2}}{a}$

Therefore, $d_{1} = 2 a \sin \frac {\theta}{2}$ and $d_{2} = 2 a \cos \frac {\theta}{2}$

Area of rhombus = $\frac {1}{2} \times d_{1} \times d_{2} = \frac {1}{2} \times 2 a \sin \frac {\theta}{2} \times 2 a \cos \frac {\theta}{2} = 2 a^{2} \sin \frac {\theta}{2} \cos \frac {\theta}{2}$

$= a^{2} \times \left( 2\sin \frac {\theta}{2} \cos \frac {\theta}{2} \right) = a^{2} \times \sin \theta$.

Note: $2\sin \frac {\theta}{2} \cos \frac {\theta}{2} = \sin \theta$.

Therefore, the area of a rhombus with the length of a side $a$ and one of the angles $\theta$ is $a^{2} \sin \theta$.

### Examples

Ex 1: Find the area of a rhombus of side $12 cm$ and the measure of one of the angles $30^{\circ}$.

Length of the side of a rhombus $a = 12 cm$

The measure of one of the angles of rhombus $\theta = 30^{\circ}$

Area of rhombus = $a^{2} \sin \theta = 12^{2} \times \sin 30^{\circ} = 144 \times \frac {1}{2} = 72 cm^{2}$.

## Conclusion

A rhombus is a 2D plane figure having four equal sides. The basic formula to find the area of a rhombus is using its diagonal. You can also find the area of a rhombus if its side and height are known and also if its side and one of the angle is known.

## Practice Problems

1. Calculate the area of a rhombus having diagonals equal to $6 cm$ and $8 cm$.
2. Calculate the area of a rhombus if its side is $14 cm$ and height is $10 cm$.
3. Calculate the area of a rhombus if the length of its side is $4 cm$ and one of its angles is $30^{\circ}$.
4. Calculate the length of the diagonal of a rhombus if the length of the other diagonal is $10 mm$ and the area is $100 mm^{2}$.

## FAQs

### How do you find the area of a rhombus?

There are three methods of finding the area of a rhombus
a) using diagonals
b) using side and height
c) using side and angle

The area of the rhombus using the diagonals is $\frac {d_{1}\times d_{2}}{2}$.

The area of the rhombus using the side and height is $a \times h$.

The area of the rhombus using the side and an angle is $a^{2} \sin \theta$.

### Is the area of a rhombus side square?

No, the area of a rhombus is not side square. It’s the formula to find the area of a square. The area of a rhombus is  $a^{2} \sin \theta$, where $\theta$ is one of its angles.

### Are the diagonals of a rhombus equal?

No, the diagonals of a rhombus are not equal. The diagonals bisect each other at right angles.

### Is the rhombus a parallelogram?

Yes, a rhombus is a parallelogram. In a parallelogram, opposite sides are equal and parallel and in a rhombus also, the opposite sides are parallel and equal.

### Is square a rhombus?

Yes, a square is always a rhombus, but a rhombus is not always a square. If in a rhombus, the measure of each angle is $90^{\circ}$, it’s called a square.

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