AREA OF RHOMBUS: FORMULA, SOLVED EXAMPLES & APPLICATION

Area of Rhombus: 

For a better knowledge of the area of a rhombus, we need to understand what a rhombus is first. Rhombus is a parallelogram whose all sides are equal and its diagonals bisect each other at right angles. Now that we know what a rhombus is, we can understand what area of a rhombus signifies. Its area refers to the two-dimensional space enclosed by all the four sides of it on a 2-D plane.

Formula:

For calculating the area of a rhombus, different parameters are used depending on the data available in the question. Area of rhombus is generally formulated following three cases given below:

1. When the length of its diagonals are given
2. When its base along with the height is given
3. When one of its sides and an interior angle is given

Derivation:

Area of rhombus when:

1. The lengths of its diagonals are given:

Let ABCD be a rhombus with the length of the diagonals as 'd_{1}' \& 'd_{2}'

As mentioned earlier in the definition, the diagonals of a rhombus bisect each other at right angles. 

From the given figure, we can see:

  ∆ AOB ≅ ∆ AOD through S-S-S congruence condition as, 

• DO=BO (diagonals of a rhombus bisect each other)
• AO ( Common side)
• AD=AB (sides of a rhombus are equal in length)

      Similarly, it can be proved that the ∆ AOB, ∆ AOD, ∆ BCD, ∆ DOC are congruent to each other.

Area of Rhombus ABCD

=\text { Area of }(\triangle A O B+\triangle A O D+\triangle B C D+\triangle D O C)

=4 \times \text{ Area of } \triangle A O B          (Since all four triangles are congruent)

=4 \times(1 / 2) \times A O \times B O sq.units

=4 \times(1 / 2) \times(1 / 2) d_1 (1 / 2) d_2 sq. units    (area of right-triangle = 1/2 x base x height)

=4 \times(1 / 8) \mathrm{d}_{\mathrm{l}} \times \mathrm{d}_{2}\text { square units} =1 / 2 \times d_1 \times d_2

\text { Area }=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}

2. The length of its base along with its height is given:

Let ABCD be a rhombus with the length of base ‘b’ and height ‘h’.

As mentioned earlier in the definition, we know that the rhombus is a parallelogram as well, so we can use the area formula of parallelogram learnt earlier to formulate the area of the rhombus. 

Area of a parallelogram = base x height   

Since rhombus is one special kind of parallelogram, this also applies to it.

\text { Area }=\text { Base } \times \text { Height }

3. The length of one of its sides and an interior angle is given:

Let ABCD be a rhombus with given side ‘a’ and interior angle ‘ѳ’. The line connecting vertex B to vertex D, BD is a diagonal of the rhombus.

 We know that the area of a triangle when two of its sides say ‘a’, ‘b’ and  included interior angle ‘Ѳ’ is calculated as follows:

Area of the triangle = ½ x a x b x sin (Ѳ) 

Now in the given figure, we can see that area of ABCD is the sum of the areas of ΔABD & the area of ΔBDC.

Area of the rhombus ABCD

= Area of ΔABD + Area of ΔBDC

= [(1 / 2) \times a \times a \times \operatorname{Sin}(\theta)]+[(1 / 2) \times a \times a  \operatorname{Sin}(\theta)]  (area of a triangle as mentioned earlier)

= a^{2} \operatorname{Sin}(\theta)

\text { Area }=a^{2} \sin \theta

Calculation Steps:

1. When the two diagonals of the rhombus are given, the area is half of the product of the two diagonals.
2. When the base and height of the rhombus is given, the product of the base and height gives the area.
3. When the angle along with the side of the rhombus is given, the product of the square of the respective side & sin of the angle gives the area enclosed.

Solved Examples:

Q1- What is the area of the rhombus whose length of the sides is 4 cm and height is 2 cm?

Ans; Since the side & height of the rhombus is given, we can use the area formula involving side & height that is:

\text { Area }=\text { Base } \times \text { Height }

Hence, Area of given rhombus = 4 cm x 2 cm = 8 \mathrm{cm}^{2}

Q2- A rhombus in which one of the angles is 30 degrees and the side is 4 cm is given. Find the area of the rhombus?

Ans: The rhombus’s angle and side are given in the question, hence we can use the Area formula: 

\text { Area }=a^{2} \sin \theta

Therefore, Area of the given rhombus =4^{2} \times \operatorname{Sin}\left(30^{\circ}\right)=16 \times(1 / 2) \mathrm{cm}^{2}=8 \mathrm{~cm}^{2}

Q3- Find the area enclosed by a rhombus with diagonals 8 cm and 6 cm?

Answer: The length of the two diagonals of the rhombus are given to be 8 cm and 6 cm. We know the formula of the area when two diagonals are given, which is”  

\text { Area }=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}

Using this we can find the area of the rhombus which is:

Area =  ½  x  8  x  6 = 24 \mathrm{cm}^{2}

Applications:

1. Automobile Industry makes use of the shape of the rhombus for designing windows.
2. It finds its use in mirrors of the vehicle.
3. The kites made are always in the shape of the rhombus.