# CLASS-8AREA OF RHOMBUS

Area of a Rhombus

Let, WXYZ is a rhombus, with the diagonals XZ and YW bisecting each other at right angles at the point O

Since WXYZ is also a parallelogram, its area

1

= 2 X area of (∆ WXY) =  2 X ------ (YW) X (OX)  =

2

1

=  (YW) X (OX) =  (YW) X ------ (XZ)    [ where OX = 1/2 XZ ]

2

1

= ------- X (YW) X (XZ) =  1/2 X product of its Diagonals

2

1

So, the area of a rhombus = ------- X product of its diagonals

2

Example.) The diagonals of a rhombus are 20 cm & 22 cm. (a) Find its area, (b) the length of a side, and (e) its perimeter.

Ans.)  Let PQRS is the given rhombus, with its diagonals PR & QS intersecting at the point O

As per the given condition, PR = 20 cm, and QS = 22 cm

1

The area of rhombus = ------ X product of the diagonals

2

= ------- X PR X QS

2

1

= ------- X 20 X 22 = 10 X 22 = 220 cm²  (Ans.)

2

(b) the diagonals of a rhombus bisects each other at right angles

1                 1

OP = ------- PR  =  ------- X 20 cm = 10 cm

2                 2

1               1

OQ = ------- QS = ------- X 22 cm = 11 cm

2               2

And ∆ POQ = 90⁰

In the right angled triangle OPQ,  PQ² = OP² + OQ²

So, PQ = √ OP² + OQ² = √10² + 11² = √100 + 121

= √221 = 14.86 = 15 cm  (Ans.)

(c) since the sides are equal, so the perimeter of the rhombus = 4 X 15 cm = 60 cm   (Ans.)