Question

A square is inscribed in a circle. If the area of the shaded region is $224\,c{m^2}$, calculate the radius.

Answer 1

Hint: Area of the shaded region will be the difference between the area of the circle and the area of a square. Square has equal sides. Consider the radius to be $r$. Then use Pythagoras theorem to find the side of the square in terms of radius. Then use the given area of the shaded region to find the radius.

Complete Step by Step Solution:
It is given that the area of the shaded region is $A = 224\,c{m^2}$

We have to calculate the radius.
We know that the area of the shaded region= Area of circle – Area of Square.
So, we have to first find the area of the circle and the area of the square.
Then, draw a diagonal of squares.
By observing the above figure, we can say that the diagonal of a square is the diameter of the circle.
So, Diagonal of square = Diameter of circle.
Let the radius of the circle be $r$
Then the diagonal of the square be $2r$
Now, let the side of the square be $a$. We know that the sides of the square are equal. And we know that the diagonal of a square divides the square into an isosceles right angled triangle. Therefore, we can use Pythagoras theorem to write,
${\left( {2r} \right)^2} = {a^2} + {a^2}$
$ \Rightarrow 4{r^2} = 2{a^2}$
By rearranging it, we can write
$ \Rightarrow {a^2} = 2{r^2}$
$ \Rightarrow a = r\sqrt 2 $
Now, using the formula,
Area of the shaded region= Area of circle – Area of Square.
We can write,
$ \Rightarrow 224 = \pi {r^2} - {(a)^2}$
Substituting the values, we got, we can write
$ \Rightarrow 224 = \pi {r^2} - 2{r^2}$
By taking common terms out and rearranging, we get
$ \Rightarrow {r^2}\left( {\dfrac{{22}}{7} - 2} \right) = 224$
By cross multiplication and further simplifying, we can write
$ \Rightarrow {r^2}\left( {\dfrac{{22 - 14}}{7}} \right) = 224$
$ \Rightarrow {r^2}\left( {\dfrac{8}{7}} \right) = 224$
$ \Rightarrow {r^2} = 224 \times \dfrac{7}{8}$
$ \Rightarrow {r^2} = 196$
$ \Rightarrow r = 14$

Hence, radius of circle is $14\,cm$

Note:
You should remember this concept that the area of a shaded region is equal to the difference between the total area of the region and the area of the un-shaded region. This concept helps in most of such questions. The diagonal of the square will be the diameter of the circle because the angle of the square is ${90^{\circ}}$. And we have a property that a circle subtends the angle of ${90^{\circ}}$ by the diameter.