Question : The area of a circle is the same as the area of a square. What is the ratio of the diameter of the circle and the diagonal of the square?
Option 1: $1:\sqrt{\pi }$
Option 2: $2:\sqrt{\pi }$
Option 3: $\sqrt{2}:\sqrt{\pi }$
Option 4: $1:{\pi }$
Correct Answer: $\sqrt{2}:\sqrt{\pi }$
Solution :
Let the area of a circle $=\pi r^2$ where $r$ is the radius of the circle.
The diameter of the circle is $2r$.
Let the area of a square $=a^2$ where $a$ is the side length of the square.
The diagonal ($d$) of the square $=a\sqrt{2}$
Given that the area of the circle is equal to the area of the square.
$\pi r^2 = a^2$
⇒ $a = r\sqrt{\pi}$
Substituting this into the formula for the diagonal of the square.
$d = a\sqrt{2} = r\sqrt{2\pi}$
The required ratio $=\frac{2r}{r\sqrt{2\pi}} = \frac{2}{\sqrt{2\pi}} = \frac{\sqrt{2}}{\sqrt{\pi}}$
Hence, the correct answer is $\sqrt{2}:\sqrt{\pi }$.
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