Question

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

Solution

## $$\Rightarrow$$  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 $$\triangle BCD,\,BC^2+CD^2=BD^2$$$$\Rightarrow$$  $$BC^2+BC^2=(2r)^2$$              [ Sides of squares are equal ]$$\Rightarrow$$  $$2BC^2=4r^2$$$$\therefore$$  $$BC^2=2r^2$$$$\Rightarrow$$  Required ratio $$=\dfrac{Area\,of\,circle}{Area\,of\,square}=\dfrac{\pi r^2}{BC^2}=\dfrac{\pi r^2}{2r^2}=\dfrac{\pi}{2}$$$$\therefore$$  Required ratio $$=\dfrac{\pi}{2}$$MathematicsRS AgarwalStandard X

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