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Find the area of the shaded portion of the semicircular f...

Find the area of the shaded portion of the semicircular figure.

Find the area of the shaded portion of the semicircular figure.

A.

$\frac{r^2}{4}(4 \pi - 3 \sqrt{3})$

B.

$\frac{r^2}{4}(2 \pi - 3 \sqrt{3})$

C.

$\frac{1}{2}r^2 \pi$

D.

$\frac{1}{8}r^2 \sqrt{3}$

E.

$\frac{r^2}{4}(4 \pi - 3 \sqrt{3})$

View answer & explanation

Correct answer is B

A_sector = $\frac{60}{360} \times \pi r^2$

= $\frac{1}{6} \pi r^2$

A_{$\bigtriangleup$} = $\frac{1}{2}r^2 \sin 60^o$

$\frac{1}{2} r^2 \times \frac{\sqrt{3}}{2} = \frac{r^2\sqrt{3}}{4}$

A_{$\text{shaded portion}$} = A_sector -
A_{$\bigtriangleup$}

= ( $\frac{1}{6} \pi r^2 - \frac{r^2\sqrt{3}}{4})^3$

= $\frac{\pi r^2}{2} - \frac{3r^2\sqrt{3}}{4}$

= $\frac{r^2}{4}(2 \pi - 3 \sqrt{3})$

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