Question

The area of a rhombus whose diagonals lengths are 16cm and 24cm is?

Answer 1

Hint: We start solving the problem by assigning variables to the length of the diagonals of the given rhombus. We then use the fact that the area of the rhombus is equal to half of the product of the lengths of the diagonals i.e., $\dfrac{{{d}_{1}}\times {{d}_{2}}}{2}$. We then substitute the given values of the lengths of the diagonals and make necessary calculations to get the required area.

Complete step by step answer:
We know that the area of a rhombus can be defined as the amount of space enclosed by a rhombus in a two – dimensional space.

Let us consider a rhombus as shown in the figure. Let ${{d}_{1}}$ be the length of the first diagonal and ${{d}_{2}}$ be the length of the second diagonal of the rhombus.
We have been given the diagonals of the rhombus as 16cm and 24cm. Let us consider ${{d}_{1}}=16cm$and ${{d}_{2}}=24cm$.
We know that if two diagonals of a rhombus are given, then the area can be found using the formula,
Area of rhombus = $\dfrac{1}{2}\times \text{product of lengths of two diagonals}$.
Area of rhombus = $\dfrac{1}{2}\times {{d}_{1}}\times {{d}_{2}}$.
Area of rhombus = $\dfrac{{{d}_{1}}\times {{d}_{2}}}{2}$.
Now, we substitute the lengths of diagonals to find the area of the given rhombus.
Area of rhombus = $\dfrac{16\times 24}{2}$.
Area of rhombus = $16\times 12$.
Area of rhombus = $192c{{m}^{2}}$.
So, we got the area of a rhombus as $192c{{m}^{2}}$.
∴ The area of the rhombus with diagonals lengths 16cm and 24cm is $192c{{m}^{2}}$.
Note:
We can alternatively solve the problem as follows:
We know that the diagonals bisect each other at an angle of ${{90}^{\circ }}$. From the figure we can see that the rhombus is divided into 4 equal right-angled triangles with length of height $\dfrac{{{d}_{1}}}{2}$ and base $\dfrac{{{d}_{2}}}{2}$.
We know that the area of the triangle is $A=\dfrac{1}{2}\times \text{base}\times \text{height}$.
Now, $A=\dfrac{1}{2}\times \dfrac{16}{2}\times \dfrac{24}{2}$.
$A=48c{{m}^{2}}$.
So, we have the area of the rhombus as 4 times the area A.
So, the area of the rhombus is $4A=4\times 48=192c{{m}^{2}}$.
We can also find the length of the sides of the rhombus using the lengths of diagonals.