Question

$AD$ is the diameter of a circle of radius $6cm$ and $AB=BC=CD$. Semi-circles are drawn with $AB$ and $BD$ as diameters as shown in the figure. Find the perimeter and area of the shaded region.

Answer 1

We have $AD=$ diameter of circle of radius $6$cm$=12$cm

So, $AB=BC=CD=\frac{12}{3}=4$cm

$BD=BC+CD=4+4=8$cm

Hence required perimeter$=$arc of semicircle of radius $6$cm$+$ arc of semicircle of radius $4$cm$+$ arc of semicircle of radius $2$cm

$=(\pi \times 6+\pi \times 4+\pi \times 2)$cm

$=12\pi$cm.

Required Area $=$ area of semicircle with $AD$ as diameter$+$area of semicircle with $AB$ as diameter$-$area of semicircle with $BD$ as diameter.

$=\frac{1}{2}\times \frac{22}{7}\times 6^2+\frac{1}{2}\times \frac{22}{7}\times 2^2-\frac{1}{2}\times \frac{22}{7}\times 4^2$

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

$=\frac{264}{7}{cm}^2$

$=37.71{cm}^2$