Question

The area of the square inscribed in a semicircle is $2c{m}^2$ . If the semi-circle is extended to a full circle, the area of the square inscribed in this circle is ___$c{m}^2$.

Answer 1

Area of square = $2c{m}^2$
Side of square = $\sqrt{2}cm$
Let r be the radius of the semicircle.
Then from the figure,
$r^2=(\sqrt{2}{)}^2+{\left(\frac{\sqrt{2}}{2}\right)}^2\Rightarrow r^2=2+\left(\frac{2}{4}\right)\Rightarrow r^2=\frac{5}{2}\Rightarrow r=\sqrt{\left(\frac{5}{2}\right)}$

When the semicircle is extended to form a full circle, the diagonal of the square inscribed in it will be the diameter of the circle.
$r=\sqrt{\left(\frac{5}{2}\right)}So,diameter=2\times \sqrt{\left(\frac{5}{2}\right)}Diagonalofsquare=side\times \sqrt{2}\Rightarrow side\times \sqrt{2}=2\times \sqrt{\left(\frac{5}{2}\right)}\Rightarrow side\times 2=2\times \sqrt{5}\Rightarrow side=\sqrt{5}$ Area of square = $sid{e}^2$
$\Rightarrow Area=(\sqrt{5}{)}^2=5c{m}^2$