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In the given figure, ABCD is a rectangle of dimensions $21\ cm \times 14\ cm$. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure.
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Given: Dimensions of rectangle $ABCD=21\ cm$\times $14 cm\ $. Diameter of semi-circle$=BC$.

To do: To find the area and perimeter of the shaded region.

Solution:

$\because$ BC is the diameter of the drawn circle $=BC=14\ cm$

$\therefore$ radius of the semi-circle, $r=\frac{14}{2} =7\ cm$

Area of shaded region $=$ Area of rectangle$-$Area of semicircle

$=21\times 14-\ \frac{\pi r^{2}}{2}$

$=294-77$

$=217\ cm^{2}$

Perimeter of shaded region $=\ AB\ +\ AD\ +\ CD\ +\ length\ of\ arc\ BC$

$=21+14+21+\pi r\ \ \ \ \ \ \ \ \ \ \ ( \because length\ of\ the\ arc\ =perimeter\ of\ the\ semi\ circle\ with\ diameter\ BC)$

$=56+\frac{22}{7} \times 7$

$=56+22$

$=78\ cm$

Thus the area of the shaded region is $217\ cm^{2}$ and the perimeter of the shaded region is $78\ cm$.

 

Updated on: 10-Oct-2022

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