Question

In the figure, a square of diagonal $8\,cm$ is inscribed in a circle. Find the area of the shaded region.

Answer 1

Hint: In this question we have been given the diagonal of the square. We know that the diagonal of the square is equal to the diameter of the circle.So we can calculate the radius of the circle by the formula $\dfrac{{diameter}}{2}$. We have to find the area of the shaded region. In the above figure, we can calculate the area of the shaded region by subtracting the area of square from the area of circle, i.e.Area of shaded region $ = $ area of circle $ - $ area of square.


Complete step by step answer:
Let us represent the data given in the figure in the diagram:

Here let us say that ABCD is a square and AC is the diagonal of the square which is also the diameter of the circle i.e.
$AC = 8\,cm$
We know the formula that if we have been given the diagonal of square, then we can calculate the area of the square by
$\dfrac{{{{\left( {diagonal} \right)}^2}}}{2}$
By putting the value in the formula we have $\dfrac{{{8^2}}}{2}$.
Simplifying it gives us $\dfrac{{64}}{2} = 32$.
So the area of the square is $32\,c{m^2}$.

Now we will calculate the area of the circle, the formula is $\pi {r^2}$ , where r is the radius.We have the diameter of the circle,
$AC = 8\,cm$
So the radius is
$\dfrac{{diameter}}{2} = \dfrac{8}{2}$
It gives us a radius of $4\,cm$.By substituting the value in the area of the circle, we have $\pi {(4)^2} = 16\pi \,c{m^2}$ .
Thus Area of shaded region $ = $ area of circle $ - $ area of square.
By putting the values we have :
 $(16\pi - 32)c{m^2}$
We can take the common factor out, and it gives $16(\pi - 2)\,c{m^2}$.

Hence the required area of the shaded region is $16(\pi - 2)\,c{m^2}$.

Note:We should note that we can calculate the area of the square by another method also. We know the formula of diagonal of square i.e.
$d = a\sqrt 2 $, where a is the side of the square.
So by putting the value of diagonal we can write
$8 = a\sqrt 2 $
On simplifying it gives us idea of the square i.e.
$a = \dfrac{8}{{\sqrt 2 }}$
Now we know the area of square is
${(side)^2} = {a^2}$
So by putting the value of side in the formula we can write
$A = {\left( {\dfrac{8}{{\sqrt 2 }}} \right)^2}$
On squaring and solving it gives us area of the square :
$\dfrac{{64}}{2} = 32\,c{m^2}$ .
We put the value of $\pi $ also in the solution i.e. $\pi = 3.14$ .
So by putting this we can write $16(3.14 - 2)$ .
On simplifying the value we can write $16(1.14)$.
It gives us the area of the shaded region $18.24\,c{m^2}$ .