Question

Find the area of the shaded region in Fig. 8, where \\

\[\overbrace{APD}\],\[\overbrace{AQB}\],\[\overbrace{BRC}\] and \[\overbrace{CSD}\] are semi-circles of diameter 14 cm, 3.5 cm, 7 cm and 3.5 cm respectively.\[\left[ Use \pi = \frac{22}{7} \right]\]

Answer 1

Area of the shaded region

= Area of the semicircle with diameter AD + Area of the semicircle with diameter BC
    − Area of the semicircle with diameter AB − Area of the semicircle with diameter CD

\[= \frac{1}{2}\pi \left( \frac{14}{2} \right)^2 + \frac{1}{2}\pi \left( \frac{7}{2} \right)^2 - \frac{1}{2}\pi \left( \frac{3 . 5}{2} \right)^2 - \frac{1}{2}\pi \left( \frac{3 . 5}{2} \right)^2 \]
\[ = \frac{1}{2} \times \frac{22}{7} \times \left( \frac{3 . 5}{2} \right)^2 \times \left[ 16 + 4 - 1 - 1 \right]\]
\[ = \frac{1}{2} \times \frac{22}{7} \times \frac{3 . 5}{2} \times \frac{3 . 5}{2} \times 18\]
\[ = 86 . 625 {cm}^2\]