Question

In the given figure, find the area of the shaded region.

     

Answer 1

Hint: First find out the area of the circle and then find the area of the rectangle inscribed inside the circle and subtract the area of rectangle from the circle to find the area of the shaded region.

Complete step-by-step answer:
First we will find the area of the rectangle = $length(l) \times breadth(b)$
Here, we are given l = 8cm and b = 6 cm
Area of rectangle = $8 \times 6 = 48 c{m^2}$
Now, we will find the area of circle = $\pi {r^2}$ (Here r is the radius of the circle)
Let us find the radius of the circle as follows:
Let us draw the diagonal of the rectangle and join BD and let it be x cm.

          

Now, since BCD is a right angled triangle, we can apply Pythagoras theorem on it.
Pythagoras theorem states: “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“
Numerically it can be written as: $${H^2} = {P^2} + {B^2}$$
Where H is the hypotenuse , P is the perpendicular and B is the base of the triangle.
Here, BD is the hypotenuse, BC is the perpendicular side and CD is the base of the triangle.
We put the values ,
$${x^2} = {6^2} + {8^2}$$
$${x^2} = 36 + 64 = 100$$
$$x = 10$$cm
Now, BD = 10 cm and since it is a rectangle, So, BD = OB+OD
OB = 5 cm
OB is also the radius of the circle. So, OB = 5 cm.
Now, area of circle = $\pi {r^2}$
$\eqalign{
  & \dfrac{{22}}{7} \times {(5)^2} \cr
  & \dfrac{{22}}{7} \times 25 = \dfrac{{550}}{7} = 78.5 c{m^2} \cr} $
Area of shaded region = area of circle – area of triangle
                                         = $78.5 c{m^2}$- $48 c{m^2}$= $30.5 c{m^2}$

Note: Find out the diagonal of the rectangle to find the diameter of the circle and we know that radius is the half of the diameter. Find out the area of both the circle and rectangle and subtract them to get the desired result.