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Area of rhombus

Objective

To show that the area of rhombus is half the product of its diagonals.

Theory

  1. A rhombus is a simple (non-self-intersecting) quadrilateral whose all four sides are of same length.
  2. If a parallelogram has two consecutive sides congruent, it is a rhombus.
  3. If two triangles are congruent then their areas are equal.
  4. Area of a triangle = 1/2 X base X height
  5. Area of a rectangle = Length X Breadth
  6. Diagonals of rhombus are perpendicular to each other.

Proof

In above figure EHGF is rhombus with diagonal HF (length d1) and diagonal EG (length d2)

Area of rhombus EHGF = Area of triangle EFH + Area of triangle FHG
                      small = frac{1}{2} 	imes EO 	imes HF + frac{1}{2} 	imes HF 	imes OG

                      small =frac{1}{2} 	imes (frac{d2}{2}) 	imes d1+frac{1}{2} 	imes (frac{d2}{2}) 	imes d1

                      small =frac{(d1	imes d2)}{4}+frac{(d1	imes d2)}{4}
                      small =frac{2(d1 	imes d2))}{4}

                      small =frac{d1 	imes d2}{2}
                      = half of the product of the diagonals

Example

Find the area of the following rhombus.

Solution:

In the given figure,

PR = d1= 24 cm.

SQ = d2 = 18 cm.

 

So, the area of the rhombus PQRS is 216 cm2.