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Example 8: Find the area of a rhombus whose diagonals are 12 cm and 18 cm.

Solution: Given that the diagonals are 12 cm and 18 cm. That is, d = 12 cm and d = 18 cm
1
2
1 1
We have, Area of a rhombus = × d × d = × 12 cm × 18 cm = 108 sq. cm
2
1
2 2
Thus, the area of the given rhombus is 108 sq. cm.
2
Example 9: The area of a rhombus is 480 cm and one of the diagonals is 30 cm. Find the other
diagonal.
2
Solution: Given that area of the rhombus = 480 cm , and one of the diagonals, d = 30 cm.
1
Let the other diagonal be d .
2
1 Quick Check
We have, area of a rhombus = × d × d 2
1
2 Find the area of a rhombus whose side
1 is 6 cm and whose altitude is 4 cm. If
\ 480 cm = × 30 cm × d 2
2
2 one of its diagonals is 8 cm long, find
d = 32 cm the length of the other diagonal.
2
Therefore, the length of the other diagonal is 32 cm.

Example 10: The diagonals of a rhombus are 18 cm and 24 cm. Find its area and the length of the
side of the rhombus. D C
Solution: Given the lengths of diagonals of the rhombus ABCD are
AC = d = 18 cm, and BD = d = 24 cm O
2
1
1 1
We have, area of rhombus = × d × d = × 18 cm × 24 cm = 216 cm 2 9 cm 12 cm
1
2
2 2
Now, to find the side of a rhombus, say AB, we use Pythagoras theorem.
A B
In right-angled triangle AOB,
1
1
2
2
AB = AO + BO 2 [Q AO = AC = × 18 cm = 9 cm, and
2 2
1 1
2
2
= 9 + 12 BO = BD = × 24 cm = 12 cm]
2
2
= 81 + 144 = 225
\ AB = 225 = 15 cm
Thus, the side of the rhombus is 15 cm.

Enrichment
Let PRST be a rhombus, and PS = d and RT = d are two diagonals of the rhombus, then
1
2
2
2
2
PR = PQ + QR (Using Pythagoras Theorem) T S
 d  2  d  2
2
PR = PQ +QR=   2   +   2   Q
2
1
2
d 2 d 2 1 P R
= 1 + 2 = d + d 2
2
4 4 2 1 2
1
Therefore, side of a rhombus = × sum of the squares of diagonals
2
237 Mensuration