Question

If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.

Answer 1

$\Rightarrow$ Let $ABCD$ be the square inscribed in the circle.

$\Rightarrow$ Let $r$ be the radius, then $2r$ will be the diameter of a circle.

$\Rightarrow$ Since square $ABCD$ is inscribed in a circle, then both diagonals are equal.

$\therefore$ $AC=BD=2r$

$\Rightarrow$ In $△BCD,B{C}^2+C{D}^2=B{D}^2$

$\Rightarrow$ $B{C}^2+B{C}^2=(2r{)}^2$ [ Sides of squares are equal ]

$\Rightarrow$ $2B{C}^2=4r^2$

$\therefore$ $B{C}^2=2r^2$

$\Rightarrow$ Required ratio $=\frac{Areaofcircle}{Areaofsquare}=\frac{\pi r^2}{B{C}^2}=\frac{\pi r^2}{2r^2}=\frac{\pi }{2}$

$\therefore$ Required ratio $=\frac{\pi }{2}$