In the given figure, four identical semicircles are drawn in quadrant. XA = 7 cm. What is the area (in cm²) of shaded region?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Area of quadrant XDTB}} \cr & = \frac{1}{4}\pi \times 14 \times 14 \cr & = \frac{1}{4} \times \frac{{22}}{7} \times 14 \times 14 \cr & = 154{\text{ c}}{{\text{m}}^2} \cr & {\text{Area of four semi - circle}} \cr & = 4 \times \frac{1}{2}\left[ {\pi \times \frac{7}{2} \times \frac{7}{2}} \right] \cr & = 2 \times \frac{{22}}{7} \times \frac{7}{2} \times \frac{7}{2} \cr & = 77{\text{ c}}{{\text{m}}^2} \cr} $$

$$\eqalign{ & {\text{Area of shaded region = 2}} \times {\text{Area of quadrilateral MONX}} \cr & = 2 \times \left[ {\frac{1}{4}\pi \times 3.5 \times 3.5 - \frac{1}{2} \times 3.5 \times 3.5} \right] \cr & = {\left( {3.5} \right)^2}\left[ {\frac{{11}}{7} - 1} \right] \cr & = \frac{4}{7} \times \frac{7}{2} \times \frac{7}{2} \cr & = 7{\text{ c}}{{\text{m}}^2} \cr} $$