Question

O is the centre of a circle of diameter 4 cm and OABC is a square, if the shaded area is $\dfrac{1}{3}$ area of the square, then the side of the square is ________.

Answer 1

Hint: Area of sector of circle with central angle $\theta$ is $\pi r^2 \times \dfrac{\theta}{360^o}$, where r is the radius of the circle.

Complete step-by-step answer:
O is the centre of the given circle and the diameter of the circle is given to be 4 cm.
The radius of the circle is half the diameter of the circle, thus, the radius of the circle is $\dfrac{1}{2} \times 4 =2$ cm.
Also, Area of the shaded region is $\dfrac{1}{3}rd$ of the area of the square.
Since OABC is a square, thus, $\angle AOC=90^o$ (All the angles of a square have measure of $90^o$).
Area of shaded region = Area of sector of circle with central angle $90^o$ = $\pi r^2 \times \dfrac{90^o}{360^o}$, where r is the radius of the given circle.
Area of shaded region = $\pi \times 2 \times 2 \times \dfrac{90}{360}$
Area of shaded region = $\pi \times 2 \times 2 \times \dfrac{1}{4}=\pi \ cm^2$
Now, let the side of the square be a cm.
Then, Area of square = $a^2$
Using the given condition that area of shaded region is $\dfrac{1}{3}rd$ of the area of the square, we get,
$a^2 \times \dfrac{1}{3}=\pi$
$\implies a^2=3\pi$
$\implies a=\sqrt{3\pi}$
Hence, the side of the square is $\sqrt{3\pi}$cm.

Note: In this type of questions, firstly we form an equation by using the given condition and then simplify the equation by using all the values given to us. After obtaining a simplified equation, we can then obtain the required value.