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}))
Correct Answer: Option B
Explanation

Asector = (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^2sqrt{3}}{4})
A(text{shaded portion}) = Asector –
A(bigtriangleup)
= ((frac{1}{6} pi r^2 – frac{r^2sqrt{3}}{4})^3)
= (frac{pi r^2}{2} – frac{3r^2sqrt{3}}{4})
= (frac{r^2}{4}(2 pi – 3 sqrt{3}))

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