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 $\frac{r^{2}}{2} (π - 2)$

$r a d i u s o f t h e c i r c u l a r = r$
$\Rightarrow A r e a o f t h e s e m i c i r c l e = \frac{π r^{2}}{2}$
$I n Δ A B C$
$C A = 2 r$
$O B = r$
$\Rightarrow A r e a o f Δ A B C = \frac{1}{2} (C A) (O B) = \frac{1}{2} (2 r) (r) = r^{2}$
$A r e a o f t h e s h a d e d r e g i o n = A r e a o f s e m i - c i r c l e - A r e a o f Δ A B C$
$= \frac{π r^{2}}{2} - r^{2} = r^{2} (\frac{π}{2} - 1)$ = $\frac{r^{2}}{2} (π - 2)$

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