Question

A square is inscribed in a circle of area $2\pi {\text{unit}}^2,$ as shown in figure. Another circle is inscribed in the square. Find the area of the shaded region.

• A. $(4-\pi ){\text{unit}}^2$
• B. $(2-\pi ){\text{unit}}^2$

Answer 1

The correct option is A $(4-\pi ){\text{unit}}^2$

Area of outer cirlc is $2\pi {\text{unit}}^2.$
Let the radius of the larger circle and smaller circle be $R\text{units}$ and $r\text{units},$ respectively.

$\therefore \pi R^2=2\pi$
$\Rightarrow R=\sqrt{2}\text{units}$

We know that the angle subtended by the diameter to any Point on the circle is equal to ${90}^o.$
Also, each corner of the square is ${90}^o,$ and it touches the circle.

$\therefore$ The diagonal of the square will pass through the center of the circle and it is the diamenter of the larger circle.

Let the side of the square is $a\text{units}.$
From $△ABC,$
$a^2+a^2=(2R{)}^2$
$\Rightarrow 2a^2=(2\sqrt{2}{)}^2=4\times 2$
(dividing by 2)
$\Rightarrow a=2\text{units}$

The line FE is perpendicular to the side of the square.
$\therefore$ It will bisect the line at F.
$\Rightarrow AF=\frac{a}{2}=1\text{unit}$
From $△AFE,$
$A{E}^2=A{F}^2+F{E}^2$
$\Rightarrow R^2=1^2+r^2$
$\Rightarrow 2=1+r^2$
$\Rightarrow r=\sqrt{2-1}=1\text{unit}$

Area of the shaded portion
$=\text{Area of square}-\text{Area of the smaller square}$
$=a^2-\pi r^2$
$=2^2-\pi \left(1^2\right)$
$=(4-\pi ){\text{unit}}^2$