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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 $$=\dfrac{\pi}{2}$$

951736_973398_ans_fde749ca57084f798f393839736f5cdb.png

Mathematics
RS Agarwal
Standard X

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