AREA OF RHOMBUS

On the webpage, we are going to see the formula to be used to find area of rhombus and some example problems.

Definition of rhombus

A quadrilateral which is having four equal sides is called rhombus. In other words a parallelogram will become rhombus if the diagonals are perpendicular. 

Area of rhombus = (1/2) x (d₁ x d₂)

Here d₁ and d₂ are diagonals.

• Properties of rhombus:
• It has four equal sides
• Both diagonals are intersecting each other at right angle.
• Opposite angles are congruent in a rhombus.
• Sum of interior angles will be 360 degree 

Difference between square and rhombus:

Example 1:

Find the area of rhombus if the diagonal is measuring 15 cm and 10 cm. 

Solution:

Here d₁ = 15 cm d₂ = 10 cm

Area of the rhombus = (1/2) x (d₁ x d₂)

 =  (1/2) x 15 x 10

 = 15 x  5

 = 75 cm²

Example 2:

Find the area of the rhombus whose vertices are A (2,-3), B (6,5), C (-2,1) and D (-6,-7).

Solution:

To find the area of the rhombus first we have to find the length of the diagonals.

Distance between two points = √(x₂ - x₁)² + (y₂ - y₁)² 

Length of diagonal AC :

A (2,-3) C (-2,1) 

x₁ = 2 ,  y₁ = -3 ,  x₂ = -3 ,  y₂ = 1

=  √(-3 - 2)² + (1 - (-3))²  ==>  √25 + (4))² ==>  √(25 + 16) ==>  √41

Length of diagonal BD :

B (6,5) D (-6,-7) 

x₁ = 6 , y₁ = 5 , x₂ = -6 , y₂ = -7

=  √(-6 - 6)² + (-7 - 5)² ==> √(12)² + (12)²  ==> √144 + 144

=   √288  ==>  √12 x 12 x 2 ==> 2√12

Area of the rhombus = (1/2) x (d₁ x d₂)

d₁ = √41 d₂ = 2√12

 = (1/2) x √41 x 2√12

=  √41 x √12

=  √12 x 41

 =  √492

=  √2 x 2 x 3 x 41

 = 2 √123 square units    

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