Question

PQRS is a diameter of a circle of radius 6 cm. The length PQ, QR and RS are equal Semicircle are drawn on PQ and QS as diameters. Find the area of the shaded region. Find also the length of boundaries of shaded portion.

Answer 1

For semicircle with diameters PS,
Radius = 6 cm
`:.` Area of such semicircle `= (1)/(2) pi r^(2) = (1)/(2) xx (22)/(7) xx 6^(2) = (396)/(7) cm^(2)`
Diameter `PS = 6 + 6 = 12 cm`
`:. PQ = QR = RS = 4 cm`
For semicircle with diameter QS, radius = 4 cm
`:.` Its area `= (1)/(2) pi r^(2) = (1)/(2) xx (22)/(7) xx 4^(2) = (176)/(7) cm^(2)`
For semicircle with diameter PQ, radius = 2 cm
`:.` Its area `= (1)/(2) xx (22)/(7) xx 2^(2) = (44)/(7) cm^(2)`
`:.` Area of shaded region `= (396)/(7) - (176)/(7) + (44)/(7) = (264)/(7) = 37.71 cm^(2)`
Given: PS = diameter of circle
and radius (r) = 6 cm
diameter `PS = 2 xx r`
`rArr PS = 2 xx 6 " " PS = 12 cm`
`PQ = QR = RS = (12)/(3) = 4 cm` each.
Hence, required perimeters = circumference of semicircle with radius 6 cm + circumference of semicircle with radius 4 cm + circumference of semicircle with radius 2 cm.
`6 pi + 4 pi + 2pi = 12 pi cm`