Question

Find the area of the shaded region given in figure.

Answer 1

Hint: In this particular question use the concept that the area of the shaded region is the difference of the area of the bigger square and the total area of the unshaded region so use these concepts to reach the solution of the question.

Complete step by step solution:

As we see that the length and the breadth of the given figure is the same so it is the square.
Therefore, length = breadth = 14 cm
Now as we know that the area of the square is side square.
So the area of the square = ${\left( {14} \right)^2} = 196$ square cm.
So the total area of the given figure or the area of the bigger square = 196 square cm.
Now we have to find out the area of the shaded region.
Now join all the points of the inner diagram as shown in the figure by dotted lines so this makes again a square.
Let the side length of this square is a cm.
So the area of the smaller square = ${a^2}$ square cm.
All four sides of the square are surrounded by the semicircle as shown in the figure, whose diameter is the side of the smaller square.
So the diameter of the semicircle is a cm.
Now as we know that the radius of the square is half of the diameter.
Therefore, r = (a/2) cm, where, r is the radius of the semicircle.
Now the area (A) of the semicircle is given as $\dfrac{1}{2}\pi {r^2}$
So the area of the four semicircles = $4\left( {\dfrac{1}{2}\pi {r^2}} \right) = 2\pi {r^2}$
Now substitute r = a/2 we have,
So the area of the four semicircles = \[2\pi {\left( {\dfrac{a}{2}} \right)^2} = 2\pi \dfrac{{{a^2}}}{4} = \dfrac{{{a^2}\pi }}{2}\] square cm.
So the total area of the unshaded region = area of the smaller square + area of the 4 semicircles.
So the total area of the unshaded region = ${a^2} + \dfrac{{{a^2}\pi }}{2}$ square cm.......... (1)
Now from figure we can say that
$14 = 3 + \dfrac{a}{2} + a + \dfrac{a}{2} + 3$
So on simplifying we have,
$ \Rightarrow 14 = 6 + 2a$
$ \Rightarrow 2a = 14 - 6 = 8$
$ \Rightarrow a = \dfrac{8}{2} = 4$ Cm.
Now substitute this value in equation (1) we have,
So the total area of the unshaded region = ${4^2} + \dfrac{{{4^2}\pi }}{2} = 16 + 8\pi $ square cm
So the total area of the unshaded region = $16 + \left( {\dfrac{{22}}{7}} \right)8 = 41.143$ square cm.
Now the area of the shaded region = area of the bigger square – total area of the unshaded region.
Now substitute the values we have,
So the area of the shaded region = 196 – 41.413 = 154.857 square cm.
So this is the required area.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the formula of the area of square, circle and semicircle which is all stated above, it will help us a lot to solve these kinds of problems, so first find out area of the bigger square as above, then find out the area of the unshaded region as above then subtract them and simplify as above we will get the required answer.