PQRS is the diameter of a circle whose radius is $r$. The length of PQ, QR and RS are equal. Two semi-circles are formed on PQ and RS. Find the perimeter of the shaded part.
$2\pi r$
$\dfrac{4 \pi r}{3}$
$\dfrac{5 \pi r}{3}$
$\dfrac{3 \pi r}{2}$
Solution
PS = 2r
PQ = QR = RS $= \dfrac{2r}{3}$, QS $= \dfrac{4r}{3}$
Perimeter of shaded part
$=$ arc SP + arc PQ + arc QS
$= \pi r + \pi \times \dfrac{2r}{3} \times \dfrac{1}{2} + \pi \times \dfrac{2r}{3}$
$= \pi r + \dfrac{\pi r}{3} + \dfrac{2 \pi r}{3} = \pi r + \pi r = 2 \pi r$