Question

In the the following figure, PSR, RTQ and PAQ are three semicircles of diameter 10 cm, 3 cm and 7 cm respectively. Find the perimeter of shaded region. [CBSE 2014]

Answer 1

Perimeter of shaded region = Length of the arc PAQ + Length of the arc PSR + Length of the arc RTQ

$$\begin{gathered} =\frac{1}{2}\times 2\mathrm{\pi }{r}_1+\frac{1}{2}\times 2\mathrm{\pi }{r}_2+\frac{1}{2}\times 2\mathrm{\pi }{r}_3 \\ =\frac{1}{2}\times 2\mathrm{\pi }\left(\frac{7}{2}\right)+\frac{1}{2}\times 2\mathrm{\pi }\left(\frac{10}{2}\right)+\frac{1}{2}\times 2\mathrm{\pi }\left(\frac{3}{2}\right) \\ =\frac{7}{2}\mathrm{\pi }+5\mathrm{\pi }+\frac{3}{2}\mathrm{\pi } \\ =10\mathrm{\pi } \\ =31.4\mathrm{cm} \end{gathered}$$
=12×2πr1+12×2πr2+12×2πr3=12×2π(72)+12×2π(102)+12×2π(32)=72π+5π+32π=72π+32π+5π=5π+5π=10π=31.4 cm
Hence, the perimeter of shaded region is 31.4 cm.