Question

Find the area of the shaded portion as shown in the adjoining figure. The four corners are quadrants of circle and at the centre there is a circle.

Answer 1

Let the center of circle be at $O$. $AC$ is diagonal of square $ABCD$

$AC=4\sqrt{2}cm$

we have $AE=CF=1cm$ each

$EF=AC-AE-CF$

$EF=4\sqrt{2}-2$

radius of circle $=\frac{1}{2}EF=\frac{1}{2}(4\sqrt{2}-2)=2\sqrt{2}-1$

Area of shaded region $=$ area (ABCD) $-$ (area of circle $+$ $4\times$ area of each quadrant)

$=4\times 4-[\pi \times (2\sqrt{2}-1{)}^2+4\times \frac{1}{4}\pi \left(1{)}^2\right]$

$=16-\pi \left\{\right(2\sqrt{2}-1{)}^2+1\}$

$=16-\pi \{8+1-4\sqrt{2}+1\}$

$=16-(10-4\sqrt{2})\pi$

$=16-13.64439$ (Taking $\sqrt{2}=1.414,\pi =3.14$)

Area $=2.36c{m}^2$