The area of the shaded region in the figure given below is:

$r^{2} (π - 2)$

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$\frac{r^{2}}{2} (π - 2)$

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$\frac{r^{2}}{2} (π - 1)$

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$r^{2} (\frac{π}{2} - 1)$

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Solution

The correct option is B
$\frac{r^{2}}{2} (π - 2)$

$G i v e n, r a d i u s o f t h e c i r c u l a r f i g u r e = r$
$\Rightarrow A r e a o f t h e s e m i c i r c l e = \frac{π r^{2}}{2}$
$I n Δ A B C$
$C A = 2 r$
$O B = r$
$\Rightarrow A r e a o f Δ A B C = \frac{1}{2} (C A) (O B) = \frac{1}{2} (2 r) (r) = r^{2}$
Area of the shaded region = Area of semi–circle - Area of $Δ A B C$
$= \frac{π r^{2}}{2} - r^{2} = r^{2} (\frac{π}{2} - 1)$ = $\frac{r^{2}}{2} (π - 2)$

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