Question

Find the area of shaded design in Fig where ABCD is a square 10cm and semicircles are drawn with each side of the square as diameter (\[\pi = 3.14\])

Answer 1

Hint: From the given figure we can say that the area of the shaded region is equal to the area obtained by subtracting the area of the square from the area of all the semicircles.

Complete step-by-step answer:
Here it is given that the side of square ABCD = 10cm
And the area of square = \[sid{e^2}\]
So, the area of square ABCD = \[{10^2} = \,100c{m^2}\]
Now as we know that here side of square is equal to the diameter of semicircle. So, the diameter of the circle = 10cm.
And, \[radius = \;\dfrac{{diameter}}{2} = \;\dfrac{{10}}{2} = \;5cm\]
Area of circle = \[\pi {r^2}\] and semicircle is half of the circle. So, the area of the semicircle must also be half of the area of the circle = \[\dfrac{1}{2}\pi {r^2}\].
Now the diameter of all the four semi circles inscribed in the square have the same diameter because as we know sides of the square are always equal and here the side of the square is equal to the diameter of the semicircle.
And from the above we come to know that the radius of all semicircles must also be equal and if the radius is also equal then the area of all the four semicircles is also equal.
Now area of four semicircles = 4 * area of one semicircle
So, Area of four semicircles = \[4 \times \dfrac{1}{2}\pi {r^2}\]
Putting the values of \[\pi \]and r in above equation
\[ \Rightarrow 4 \times \dfrac{1}{2} \times 3.14 \times {(5)^2}\]
\[ \Rightarrow 2 \times 3.14 \times 25 = \;157c{m^2}\]
Now as we know, Area of 4 semi circles – Area of square will give the area of shaded design.
So, now put the values of area of semicircles and area of square in the above equation.
\[ \Rightarrow 4 \times \dfrac{1}{2}\pi {r^2}\; - \;sid{e^2}\]= area of shaded design
\[ \Rightarrow \;157 - 100 = 57c{m^2}\]
Hence the area of shaded design is \[57c{m^2}\].

Note: Whenever we come up with this type of problem then we must solve these type of problem by first finding the area of inner shape ( here semicircle ) then finding the area of outer shape ( here square ) and then subtracting the lower value from the higher value we get the area of shaded region.