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.
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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 $42\ cm^2$.

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Updated on: 10-Oct-2022

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