Question

The area of the smallest circle containing a square of side $a$ is:-

Answer 1

Hint: From the given side of square, find the length of diagonal of square of side $a$ using the formula, $\text{length of diagonal}=\sqrt{2}\left(\text{length of side}\right)$. The diagonal of the square circumscribed is the diameter of the circle. Find the radius of the circle and then calculate the area of the circle, $A=\pi r^2$ where $r$ is the radius of the circle.

Complete step-by-step answer:
Given, the length of the side of square is $a$ units.
As square is circumscribed by a circle, the diagonal of the square is equal to the diameter of the circle.
We will first calculate the diagonal of the square using the formula, $\text{length of diagonal}=\sqrt{2}\left(\text{length of side}\right)$
Therefore, the diagonal of the square is $\sqrt{2}a$ units.
Also, $\sqrt{2}a$ is the diameter of the circle.
As, we know that the radius is half the diameter.
Hence, radius of the given circle is ${\displaystyle \frac{\sqrt{2}a}{2}}$ units.
Now, we will calculate the area of the circle using the formula, $A=\pi r^2$, where $r$ is the radius of the circle.
On substituting the value of the radius ${\displaystyle \frac{\sqrt{2}a}{2}}$ units in the formula for area, $A=\pi r^2$ we get
$$\begin{gathered} A=\pi {\left({\displaystyle \frac{\sqrt{2}a}{2}}\right)}^2 \\ A=\pi {\displaystyle \frac{2a^2}{4}} \\ A={\displaystyle \frac{\pi a^2}{2}} \end{gathered}$$
Hence, the area of the smallest circle containing a square of side $a$ is square units ${\displaystyle \frac{\pi a^2}{2}}$

Note:- The key point to solve this question is that the diagonal of the square circumscribed in the circle is equal to the diameter of the circle. After finding the radius of the circle, find the area of the circle using the formula, $A=\pi r^2$ where $r$ is the radius of the circle.