Question

In given figure, OE $=20$ cm. In sector OFST, square OEFG is inscribed. Find the area of the shaded region.

Answer 1

$\Rightarrow$ $OE=OG=GF=FE=20cm$ [ Sides of squares ]

$\Rightarrow$ In right angled $△OEF,(OF{)}^2=(OE{)}^2+(EF{)}^2$

$\Rightarrow$ $(OF{)}^2=(20{)}^2+(20{)}^2$

$\Rightarrow$ $(OF{)}^2=400+400$

$\Rightarrow$ $(OF{)}^2=800$

$\therefore$ $OF=20\sqrt{2}cm$

$\therefore$ $OF=r=20\sqrt{2}cm$

$\Rightarrow$ Area of sector $OSFT=\frac{\theta }{{360}^o}\pi r^2$

here $\theta ={90}^o$

$\Rightarrow$ Area of sector $OSFT=\frac{{90}^o}{{360}^o}\times 3.14\times (20\sqrt{2}{)}^2$

$\Rightarrow$ Area of sector $OSFT=\frac{3.14\times 800}{4}=628c{m}^2$

$\Rightarrow$ Let $a=20cm$

$\Rightarrow$ Area of square $OEFG=a^2=(20{)}^2=400c{m}^2$

$\Rightarrow$ Area of shaded region = Area of sector $OSFT$ - Area of square $OEFG$

$\therefore$ Area of shaded region $=628-400=228c{m}^2$