Question : The area of a circle whose radius is the diagonal of a square whose area is $4\;\mathrm{cm^2}$ is:
Option 1: $16\pi\;\mathrm{cm^2}$
Option 2: $4\pi\;\mathrm{cm^2}$
Option 3: $6\pi\;\mathrm{cm^2}$
Option 4: $8\pi\;\mathrm{cm^2}$
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Correct Answer: $8\pi\;\mathrm{cm^2}$
Solution :
The area of a square $= s^2$, (where $s$ is the side length of the square.)
Given that the area of the square $=4\;\mathrm{cm^2}$
The side length of the square $=\sqrt{4} = 2\;\mathrm{cm}$
$\therefore$ The diagonal of a square $ = s\sqrt{2}=2\sqrt{2}\;\mathrm{cm}$
This diagonal is the radius of the circle.
The area of a circle $= \pi r^2$, (where $r$ is the radius of the circle.)
$\therefore$ The area of the circle $= \pi (2\sqrt{2})^2 = 8\pi\;\mathrm{cm^2}$
Hence, the correct answer is $8\pi\;\mathrm{cm^2}$.
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