Question

What is the area of the circle that can be inscribed in a square of side $6cm$?

Answer 1

Hint: In this question we have been asked to find the area of the circle inscribed in a square of side $6cm$. We will solve this question by using the relation that the length of the side of the square is equal to the diameter of the inscribed circle. We will then find the radius of the circle by using the relation that $d=2r$ where $d$ is the diameter of the circle and $r$ is the radius of the circle. We then find the area of the circle by using the formula $\pi {{r}^{2}}$ and get the required solution.

Complete step by step solution:
We know that the length of the side of the square is $6cm$. we have a circle inscribed in the square therefore, we can see the diagram as:

Now the relation that the length of the side of the square is equal to the diameter of the inscribed circle therefore, we get:
$\Rightarrow d=6cm$
Now we know that relationship that $d=2r$ where $d$ is the diameter of the circle and $r$ is the radius of the circle therefore, we get:
$\Rightarrow 2r=6cm$
On rearranging, we get:
$\Rightarrow r=\dfrac{6}{2}cm$
On simplifying, we get:
$\Rightarrow r=3cm$, which is the required radius.
Now we know that the area of the circle is given by the formula $A=\pi {{r}^{2}}$therefore, we get:
$\Rightarrow A=\pi \times 3\times 3\text{ }c{{m}^{2}}$
On multiplying, we get:
$\Rightarrow A=9\pi \text{ }c{{m}^{2}}$, which is the required solution.

Note: It is to be noted that an inscribed shape is the largest possible shape that can be made inside a plane figure. This means that the circle should touch all the $4$ sides of the square when it is to be inscribed in a square. A circle can also be inscribed in other shapes such as a triangle or a polygon.