Question

$PQRS$ is a diameter of a circle. The lengths of $PQ,QR$ and $RS$ are equal. Semicircles are drawn on $PQ$ and $QS$ to create the shaded figure. What is the perimeter of the shaded figure?

• A. $2\pi r$
• B. $\pi r$
• C. $2\pi r$
• D. $3r$

Answer 1

The correct option is A $2\pi r$

$PS=2r\Rightarrow PQ=QR=RS=\frac{2r}{3}$
$\therefore$ Perimeter of the shaded portion $=$ Smaller arc PQ $+$ Arc QS $+$ Arc PS
$=\frac{1}{2}(2\pi \times \frac{r}{3})+\frac{1}{2}(2\pi \times \frac{2r}{3})+\frac{1}{2}\times 2\pi r$
$=\frac{\pi r}{3}+\frac{2\pi r}{3}+\pi r$

$=2\pi r$