Question

In the given figure ABCD is a square whose sides are of length 2cm. The mid points of side AB, BC, CD and DA are respectively P, Q, R and S. four arcs are drawn with A, B, C and D as centres and AP, BQ, RD and SA as radius. Find the area of the shaded portion.

$\begin{align}
  & a)\left( 4-\dfrac{\pi }{3} \right)c{{m}^{2}} \\
 & b)\left( 4-\pi \right)c{{m}^{2}} \\
 & c)\left( 2-\dfrac{\pi }{2} \right)c{{m}^{2}} \\
 & d)\left( 3-\dfrac{\pi }{7} \right)c{{m}^{2}} \\
\end{align}$

Answer 1

Hint: In the figure, area of shaded portion is given by Area of square – Area of all 4 quarter circle. We know the formula for area of square is ${{\left( side \right)}^{2}}$ and area of quarter circle is given by $\dfrac{\pi {{r}^{2}}}{4}$ . Hence we will calculate all the areas and then find the area of shaded region by subtracting the areas of quarter circles from area of square.

Complete step by step answer:

Now first consider the diagram given.

In this figure ABCD is a square with each side as 2cm.
Hence we have AB = BC = CD = DA = 2cm ……………………….. (1)
Now we know that Area is square is given by ${{\left( side \right)}^{2}}$ .
Hence Area of square ABCD is ${{2}^{2}}=4c{{m}^{2}}........................\left( 2 \right)$
Now consider 4 quarter circles.
Now we have that P, Q, R and S are mid points of each side respectively. Hence from equation (1) we can say AP = BQ = RD = SA = $\dfrac{2}{2}$= 1cm.
Now these are nothing but radius of respective quarter circles.
Now area of quarter circle is given by $\dfrac{\pi {{r}^{2}}}{4}$
Now since radius of each quarter circle is equal their Area will also be equal.
Hence we get radius of each quarter circle = $\dfrac{\pi {{\left( 1 \right)}^{2}}}{4}................................\left( 3 \right)$
Now we can see that the area of shaded portion is nothing but Area of square – Area of 4 quarter circles.
Hence, from equation (2) and (3) we get.
Area of shaded region $=4-\dfrac{\pi }{4}-\dfrac{\pi }{4}-\dfrac{\pi }{4}-\dfrac{\pi }{4}$
Taking – 1 common we get
$\begin{align}
  & =4-1\left( \dfrac{\pi }{4}+\dfrac{\pi }{4}+\dfrac{\pi }{4}+\dfrac{\pi }{4} \right) \\
 & =4-\left( \dfrac{4\pi }{4} \right) \\
 & =4-\pi \\
\end{align}$
Hence area of shaded portion is $4-\pi $ .
So, the correct answer is “Option b”.

Note:
 We know that 4 quarter circles of same radius is equal to circle. Hence we can directly solve this questions by replacing area of 4 quarter circles of radius 1cm by area of circle with radius 1cm and hence the formula for shaded region would be area of square – area of circle with radius 1cm.