Question

In the following figure, length of AB and AD are given as 12 m and 5 m respectively. Find the area of the shaded region. [Use $\pi =3.14$]
 

Answer 1

Hint: As you can see the figure, given above you will find that ABCD is a rectangle with length of AB and AD as 12 m and 5 m respectively and we have to find the area of the shaded region. Area of the shaded region is the subtraction of the rectangle from the area of the circle. Now, the area of the rectangle is equal to multiplication of length and breadth. Substitute the length and breadth given in the above question. The formula for the area of a circle is equal to $\pi {{r}^{2}}$ where “r” is the radius of the circle. Radius is calculated from the half of the diagonal of this rectangle ABCD.

Complete step-by-step answer:
Area of the shaded region is equal to the subtraction of the area rectangle from the area of the circle.
Area of rectangle is calculated by the following formula:
$l\times b$
In the above expression, “l and b” is the length and breadth of the rectangle which is given in the above question as 12 m and 5 m respectively.
Substituting these values of l and b in the above formula for area of the rectangle we get,
$\begin{align}
  & 12\times 5 \\
 & =60{{m}^{2}} \\
\end{align}$
Now, we are going to find the area of the circle as follows:
We know the formula for area of the circle as:
$\pi {{r}^{2}}$
In the above expression, “r” is the radius of the circle. Now, radius is calculated by taking half of the diagonal of the rectangle ABCD.

There is a property of the circle that the angle subtended by the diameter on to any part of the circle is ${{90}^{\circ }}$ and as you can see that angle BAD is the right angle. Hence, the diagonal of the rectangle BD is the diameter of the circle.
Applying Pythagoras theorem on the triangle BAD we get,
${{\left( Hypotenuse \right)}^{2}}={{\left( Base \right)}^{2}}+{{\left( Perpendicular \right)}^{2}}$
$\begin{align}
  & {{\left( BD \right)}^{2}}={{\left( AB \right)}^{2}}+{{\left( AD \right)}^{2}} \\
 & \Rightarrow {{\left( BD \right)}^{2}}={{\left( 12 \right)}^{2}}+{{\left( 5 \right)}^{2}} \\
 & \Rightarrow {{\left( BD \right)}^{2}}=144+25 \\
 & \Rightarrow {{\left( BD \right)}^{2}}=169 \\
 & \Rightarrow BD=13m \\
\end{align}$
From the above calculations, we have got the diameter of the circle as 13 cm and we know that half of the diameter is the radius of the circle.
$\dfrac{13}{2}m$
Hence, we got the radius of the circle as $\dfrac{13}{2}m$.
Substituting the above value of radius and $\pi =3.14$ in the formula for the area of circle we get,
$\pi {{r}^{2}}$
$\begin{align}
  & \left( 3.14 \right){{\left( \dfrac{13}{2} \right)}^{2}} \\
 & =132.665{{m}^{2}} \\
\end{align}$
Hence, we got the area of the circle as $132.66{{m}^{2}}$.
Now, subtracting the area of rectangle from the area of the circle we get,
$\begin{align}
  & 132.66-60 \\
 & =72.66{{m}^{2}} \\
\end{align}$
Hence, we got the area of the shaded region as $72.66{{m}^{2}}$.

Note: In this question, make sure you put the units correctly corresponding to the area of the rectangle and circle and the length of the radius. There are two things to be taken care of in subjective questions, if you miss the units then it will cost you some marks and in the objective paper, examiner generally, play with the units like in this problem you are lucky that units of both the lengths of the rectangle are in “m” but sometimes you will find that one of the length is in “m” and other in “cm” and you have given the area in ${{m}^{2}}$ so you have to take care of the units.