Question

$PQRS$ is a diameter of a circle of radius $6cm$. The lengths $PQ,QR$ and $RS$ are equal. Semicircles are drawn on $PQ$ and $QS$ as diameters, as shown in the figure. Find the perimeter of the shaded region.

Answer 1

We have, $PS=$ Diameter of a circle of radius $6m=12cm$

$\therefore PQ=QR=RS=\frac{12}{3}=4cm$

$QS=QR+RS=(4+4)cm=8cm$

Hence required perimeter

= Area of semicircle of radius $6cm+$ Arc of semicircle of radius $4cm+$ ARC of semicircle of radius $2cm$

$=(\pi \times 6+\pi \times 4+\pi \times 2)cm=12\pi cm=12\times \frac{22}{7}=\frac{264}{7}=37.71cm$

Required arc = Area of semicircle with $PS$ as diameter + Area of semicircle with $PQ$ as diameter - Area of semicircle with $QS$ as diameter.

$=\frac{1}{2}\times \frac{22}{7}\times (+{)}^2+\frac{1}{2}\times \frac{22}{7}\times (4{)}^2$

$=\frac{1}{2}\times \frac{22}{7}(6^2+2^2-4^2)=\frac{1}{2}\times \frac{22}{7}\times 24=\frac{264}{7}c{m}^2$

$=37.71c{m}^2$