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Find the area of the shaded region in the given figure, if $ABCD$ is a square of side $14\ cm$ and $APD$ and $BPC$ are semicircles.
"
Given:
$ABCD$ is square of side 14 cm and $APD$ and $BPC$ are semicircles.
To do:
We have to find the area of the shaded region.
Solution:
Here, as given in the above question $ABCD$ is a square and $APD$ and $BPC$ are two semi-circles.
$\because ABCD$ is a square.
$\because$ Side of the square ABCD is 14 cm.
$\therefore AB=BC=CD=DA=14\ cm$.
Here, AD and BC are diameters of semi-circles APD and BPC.
$\therefore$ Radius of the semi-circles APD and BPC $=\frac{14}{2}\ cm=7\ cm$.
Therefore,
Area of square ABCD $=(14)^2\ cm^2=196\ cm^2$.
Area of semicircle APD$=\frac{22}{7}\times\frac{7^2}{2}\ cm^2=11\times7\ cm^2=77\ cm^2$.
Area of semicircle BPC$=\frac{22}{7}\times\frac{7^2}{2}\ cm^2=11\times7\ cm^2=77\ cm^2$.
Area of the shaded region$=$Area of the square ABCD$-$(Sum of the areas of semicircle APD and BPC)
$=196-(77+77)\ cm^2$
$=196-154\ cm^2$
$=42\ cm^2$
The area of the shaded region is $42\ cm^2$.