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Example 7: Find the area of the quadrilateral PQRS shown in the figure.

P

9.3 cm
S d = 12.5 cm Q

8.7 cm

R
Solution: Given, d = 12.5 cm, h = 9.3 cm, and h = 8.7 cm. P
1
2
1
We have, area of a quadrilateral = × d ×(h + h ) h 1 = 9.3 cm
1
2
2
1 S d = 12.5 cm Q
= × 12.5 cm × (9.3 + 8.7) cm
2 h 2 = 8.7 cm
1
= × 12.5 cm × (18) cm = 112.5 sq. cm R
2
Area of a Special Quadrilateral (Rhombus)

We know that a rhombus is a special type of
quadrilateral with the following properties: Maths Talk

(a) All four sides are equal in length and its opposite You know that a parallelogram is
sides are parallel to each other. also a quadrilateral. Make a parallelogram on
a paper sheet with proper dimensions and find
(b) Both diagonals are perpendicular bisectors of its area. Divide it into two triangles by using a
each other. pair of scissors. Find the area of triangles. Does
Since a rhombus is a special quadrilateral, the same this satisfy the formula of parallelogram that
method of splitting into triangles can be used to find you already know? Discuss in the class.
b
the area of a rhombus.

In the following figure, ABCD is a rhombus, and the h h
diagonals AC and BD bisect perpendicularly to each b
other at point O. Therefore,

Area of rhombus ABCD = Area of DADC + Area of DABC
1 1
= × AC × DO + × AC × BO D C
2 2
(Q DO and BO are perpendicular to AC)

1
= × AC × (DO + BO) O
2
1
So, area of the rhombus ABCD = × AC × BD (Q DO + OB = BD) A B
2
Since AC and BD are the diagonals, and if they are represented by d and d respectively,
1
2
1
then the area of a rhombus = × d × d , where d and d are the diagonals of the rhombus.
1
2
1
2
2
1
Thus, the area of a rhombus = × product of its diagonals
2
Mathematics-8 236