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

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

Answer: Option D

Solution

Answer: Option D

Solution:

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

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