Question

If the side of a rhombus is 3 cm and its one diagonal is 4 cm, then find the area of the rhombus.

• A. $\sqrt{5}{\text{cm}}^2$
• B. $2\sqrt{5}{\text{cm}}^2$
• C. $3\sqrt{5}{\text{cm}}^2$
• D. $4\sqrt{5}{\text{cm}}^2$

Answer 1

The correct option is D $4\sqrt{5}{\text{cm}}^2$

Side of the rhombus = 3 cm

Length of a diagonal = 4 cm

Semi-perimeter (s) of the triangle containing the diagonal,
$s=\frac{(3+3+4)}{2}$ = 5 cm

The area (A) of the triangle with sides a, b and c can be calculated using Heron's formula, given by:
$A=\sqrt{s(s-a)(s-b)(s-c)}$.

$=\sqrt{5(5-3)(5-3)(5-4)}$

$=\sqrt{5\times 2\times 2\times 1}$

$=2\sqrt{5}{\text{cm}}^2$

Therefore , area of the triangle containing the diagonal is $2\sqrt{5}{\text{cm}}^2$.
Area of rhombus
$=2\times \left(\text{Area of the triangle containing the diagonal}\right)$$=2\times 2\sqrt{5}$
$=4\sqrt{5}{\text{cm}}^2$