Question

In the figure given here $◻ABCD$ is a square of side 50 m. Points P, Q, R and S are the mid-points of side AB, BC, CD and AD respectively. Find the area of the shaded region.

Answer 1

Hint: Firstly, find the area of square $◻ABCD$ .
Then, find radii of the given quarter circles and thus find the total area of the 4 quarter circles.
Finally, subtract the area of 4 quarter circles from the area of the square to get the area of the required shaded area.

Complete step-by-step answer:
Here, $◻ABCD$ is a square of side 50 m.
So the area of square $◻ABCD$ is $ABCD={\left(50\right)}^2=2500m^2$ .
Now, there are 4 quarter circles given in the diagram with centres of the circles as points A, B, C and D and also points P, Q, R and S are the mid-points of side AB, BC, CD and AD respectively.
Thus, the length of the segment AP = PB = BQ = QC = CR = RD = DS = SA $={\displaystyle \frac{50}{2}}=25m$ .
So, the radii of the given quarter circles are 25 m each.
Now, the area of 4 quarter circles with radii 25 m each is given by $4\times {\displaystyle \frac{\pi r^2}{4}}=\pi r^2$ .
Thus, the are of given 4 quarter circles $=\pi {\left(25\right)}^2$
    $$\begin{gathered} ={\displaystyle \frac{22}{7}}\times 625 \\ =1964.28 \\ =1964m^2 \end{gathered}$$
Now, the area of the shaded part can be given by the difference of area of square and the area of 4 quarter circles.
$\therefore$ Area of shaded region $=2500-1964=536m^2$ .
Thus, we get the required area as $536m^2$ .

Note: Try to remember the following formula for solving these types of questions 1.Area of a square with length of side $a=a^2$
2.Area of a circle with radius $r=\pi r^2$
3.Area of semicircle with radius of circle $r={\displaystyle \frac{\pi r^2}{2}}$
4.Area of quarter circle with radius of circle $r={\displaystyle \frac{\pi r^2}{4}}$