Question

State true or False.
Areas of the shaded regions in figures 1 and 2 are equal. The corresponding lengths and breadths of the rectangles are equal, and the radius of the quarter circles in figure 1 is half the breadth of the rectangle.

• A. True
• B. False

Answer 1

The correct option is A True

Area of the region shaded in red in figure 1 = Area of the rectangle - Area of the four quarter circles.
Radius of each quarter circle = $\frac{BreadthofRectangle}{2}$ = $\frac{b}{2}$

Area of rectangle $=l\times b=lb$
Area of 4 quarter circles $=4\times \frac{\pi r^2}{4}$
$=\frac{\pi b^2}{4}$

Therefore, Area of region in red = Area of rectangle - Area of 4 quarter circles
$\Rightarrow lb-\frac{\pi b^2}{4}$

Therefore, Area of the region shaded in red in figure 2 = Area of the rectangle - Area of the circle inside the rectangle.
Here, observe that diameter of the circle = Breadth of the rectangle
Radius of circle = $\frac{b}{2}$

Therefore, area of region in red = Area of rectangle - Area of circle
$\Rightarrow l\times b-\pi {\left(\frac{b}{2}\right)}^2$
$\Rightarrow lb-\frac{\pi b^2}{4}$

We can see that the areas of the shaded regions are same in both the figures.