Question

In the given figure, find the area of the shaded region. [Use = 3.14]

Answer 1

To Find :

The area of the shaded region .

The Shaded region is the area left after removing a rectangle out of a circle.

Given :

• The length of rectangle = 12 cm
• the breadth of rectangle = 5 cm
• radius of rectangle = ???

Formula to be applied:

• Pythagorean Theorem:

Hypotenuse² = Hieght² + Base²

• Area of Circle

$\bold{ { \pi \: r}^{2} }$

• Area of rectangle

l×b

Construction:

• construct a line segment and diagonal BD
• This is also the diameter of the rectangle

Solution :

The solution is in 4 steps , namely

• Area of rectangle,
• Diameter of Circle,
• Area of Circle , and final step
• Area of shaded Region

1] The area of rectangle:

We have to find out the area of Rectangle to be removed from the Area of circle .

Length of the rectangle = 12 cm

breadth of the rectangle = 5 cm

$\implies \bold{area = 5 \times12}$

$\implies \boxed{ \bold{area = 60 {cm}^{2} }}$

hence area of rectangle = 60 cm²

2] Diameter of the circle :

The rectangle is situated in the middle of the Circle and the diagonals cut through the centre of the rectangle. hence the diagonal of the rectangle is the diameter of the circle .

The BD is a diameter of Circle

$\bold{ \triangle \: ABD \: is \: a \: right \: angle \: triangle \: } \\ \implies \bold{BD \:is \: the \: hypotenuse \: of \: the \triangle}$

So, using the Pythagorean Theorem, we will find out the value of BD .

$\implies \bold{ {hypotenuse}^{2} = {5}^{2} + 1 {2}^{2} }$

$\implies \bold{ {BD}^{2} = 144 + 5}$

$\implies \bold{BD = \sqrt{169}}$

$\implies \boxed{ \bold{BD = 13 \: cm}}$

so, BD = 13 cm

so, radius of circle = diameter/2

so, radius = 13/2

3] Area of circle :

To find the the area of shaded Region we have to find the area of the circle

Radius = 13/2

pi = 3.14

$\implies \bold{area = 2 \pi \: r}$

$\implies \bold{area = 2 \times 3.14 \times \frac{13}{2 } \times \frac{13}{2}}$

$\implies \bold{area = 3.14 \times 6.5 \times 13}$

$\implies \boxed{ \bold{area = 265.33 \: {cm}^{2} }}$

so, area of the circle = 265.33 cm²

4] Area of the Shaded Region :

The area if shaded region :

area of circle - Area of rectangle

$\implies \bold{265.33 \: c {m}^{2} - 60 \: c {m}^{2} }$

$\implies \boxed{ \boxed{ \bold{area \: of \: shaded \: region \: = 205.33 \: c {m}^{2} }}}$

hence, area of shaded Region = 205.33 cm²