GATE CS 2018 Q.3 (GA Q.3): Quantitative Aptitude

The area of a square is $d$ . What is the area of the circle which has the diagonal of the square as its diameter?

$\pi d$

$\pi d^2$

$\frac{1}{4} \pi d^2$

$\frac{1}{2} \pi d$

Official answer and solution

Solution

Relate the square's area to its diagonal, which serves as the circle's diameter, then compute the circle's area.

Let $s$ be the side length of the square. The area of the square is $s^2 = d$ .

The diagonal of the square is $D = s\sqrt{2}$ . Given that the circle has diameter equal to $D$ , we have $D = s\sqrt{2}$ .

The area of a circle with diameter $D$ is:

\text{Area} = \frac{\pi D^2}{4} = \frac{\pi (s\sqrt{2})^2}{4} = \frac{\pi \cdot 2s^2}{4} = \frac{\pi s^2}{2}

Substituting $s^2 = d$ gives $\frac{1}{2}\pi d$ .

Final answer

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