Question

If the area of a square is $32c{{m}^{2}}$ then find the diagonal of the square.
\[\begin{align}
  & A.8cm \\
 & B.4\sqrt{2}cm \\
 & C.8\sqrt{2}cm \\
 & D.4cm \\
\end{align}\]

Answer 1

Hint: In this question, we are given the area of a square and we need to find the diagonal. For this, we will first find the side of the square using the formula of the area of square given by, $\text{Area of the square}=\text{side}\times \text{side}={{\left( \text{side} \right)}^{2}}$. After that, we will use Pythagoras theorem to calculate the diagonal of the given by ${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{Base} \right)}^{2}}+{{\left( \text{Perpendicular} \right)}^{2}}$.

Complete step by step answer:
Here we are given the area of the square as $32c{{m}^{2}}$. We need to find the length of the diagonal of the square. Let us suppose the side of the square as s and the diagonal of the square as d. Our figure looks like this:

Now let us find the side of the square. As we know that, the area of the square is given by $A={{s}^{2}}$ so we get:
$32={{s}^{2}}\Rightarrow s=\sqrt{32}cm$.
Hence the side of the square is $\sqrt{32}cm$.
Now we need to find the value of d.
As we know, every angle in a square is ${{90}^{\circ }}$. So we can say from the figure that triangle ACD is a right angled triangle. Therefore we can use Pythagoras theorem in triangle ACD. Pythagoras theorem is given as:
${{\left( \text{Hypotenuse} \right)}^{2}}={{\left( \text{Base} \right)}^{2}}+{{\left( \text{Perpendicular} \right)}^{2}}$.
In $\Delta ACD$ we have, ${{\left( \text{AC} \right)}^{2}}={{\left( \text{CD} \right)}^{2}}+{{\left( \text{AD} \right)}^{2}}$.
From figure we can see that AC = d, AD = DC = s, so we get:
${{\left( \text{d} \right)}^{2}}={{\left( \text{s} \right)}^{2}}+{{\left( \text{s} \right)}^{2}}\Rightarrow {{\text{d}}^{2}}=2{{\text{s}}^{2}}$.
Now as found earlier the value of s is $\sqrt{32}cm$. Hence we get:
${{d}^{2}}=2{{\left( \sqrt{32} \right)}^{2}}\Rightarrow {{d}^{2}}=2\times 32\Rightarrow {{d}^{2}}=64$.
Taking square roots both sides we get:
$d=\sqrt{64}\Rightarrow d=8$.
Hence the length of the diagonal of the square is 8cm.

So, the correct answer is “Option A”.

Note: Students should always try to draw diagrams for better understanding. Make sure to use proper units with area, length. We use squared units for area. Since a square has both diagonal equal so we can take any triangle and find any diagonal. Students can also learn a shortcut formula to find diagonal of the square given as: Diagonal of the square $\Rightarrow \sqrt{2}\times \text{side}$.