Question

In given figure, $ABCD$ is a square of $14cm$, find the area of shaded portion, if all larger circles are equal

• A. $\frac{57}{2}(3-\sqrt{2})c{m}^2$
• B. $\frac{59}{3}(3-\sqrt{3})c{m}^2$
• C. $\frac{77}{2}(3-2\sqrt{2})c{m}^2$
• D. None of these

Answer 1

The correct option is C $\frac{77}{2}(3-2\sqrt{2})c{m}^2$

$AB=14cm$
Then, $C_1{C}_2=7\sqrt{2}cm$
Let $r$ is the radius of shaded circle.
Then, $7\sqrt{2}=\frac{7}{2}+\frac{7}{2}+2r$
$\Rightarrow \frac{7}{2}(\sqrt{2}-1)=r$
Area of shaded circle $=\pi r^2$
$=\pi \times {\left[\frac{7}{2}\right(\sqrt{2}-1\left)\right]}^2$
$=\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}(\sqrt{2}-1)$
$=\frac{77}{2}(3-2\sqrt{2})c{m}^2$