Question

Find the area of the blue shaded portion, where the length of the lower part is 14 units and radius of the upper semi-circle is 7 units.

(a) 196
(b) 98
(c) 200
(d) None of these

Answer 1

Hint: Let us ignore the bottom semi-circle and consider it as a part of the figure. So the area = area of rectangle + area of the semicircle above. But we do not need the semi-circle below. So area = area of rectangle + area of the semicircle above – area of semi-circle below. Now note that the area the semi-circle above = area of semi-circle below. Cancel them and find the area of the rectangle which is the final answer.

Complete step-by-step answer:
In this question, we are given the figure:

It is given that the length of the lower part is 14 units and radius of the upper semi-circle is 7 units.
We need to find the area of the blue shaded region.
In the figure, note that at the bottom, a semicircle is removed from the rectangle.
This removed semicircle has the radius same as the semi-circle above.
For a moment, let us ignore the removed semi-circle. The figure will look like the following:

So, the area of this figure will be the area of the rectangle + area of the semicircle above.
Now, we need to remove the area of the semicircle below from the equation above.
So, required area = area of rectangle + area of the semicircle above – area of semi-circle below.
Now, since the radius of both the semi circles is the same, so the area of both the semi-circles will also be the same. So, in the equation above, the second and the third terms will cancel each other out.
Using this, we will get the following:
Required area = area of rectangle
Now, we are given that the length of the rectangle = 14 units.
Notice that the width of the rectangle is equal to the diameter of the semicircle = 14 units.
Area of rectangle = length of rectangle ${\times}$ width of rectangle
So, area of rectangle = 14 ${\times}$ 14
Hence, the area of the shaded region = 196 sq. units.
Hence, option (a) is correct.

Note: In this question, it is very important to note that the radius of both the semi circles is the same, so the area of both the semi-circles will also be the same. Using this, you can directly cancel these out. This will save a lot of time as compared to actually calculating the areas of the semi-circles.