GATE CSE 2018 | Question: GA-3

Question

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

• A. $\large{\pi} d$
• B. $\large{\pi} d^2$
• C. $\dfrac{1}{4}\large{\pi} d^2$
• D. $\dfrac{1}{2}\large{\pi} d$

Comments

(D) is the answer

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Area of square $= (side)^2=d$

$side=\sqrt d$

Length of diagonal (square) = $\sqrt2\ side$

$\sqrt2\ side = diameter$

$\dfrac{\sqrt2\ side}{2}=radius$

Area of circle = $\pi r^2=\pi \dfrac{\sqrt2\ side}{2}\dfrac{\sqrt2\ side}{2}=\pi \dfrac{\sqrt2\ \sqrt d}{2}\dfrac{\sqrt2\ \sqrt d}{2}=\dfrac{\pi d}{2} $

Answer 1

The area of a square is $d$

We know that, if a square has $a$ units as the length of its side, then its area will be $a^2.$

 

$\Rightarrow \textbf{every side of the square will be } \sqrt{d}$

Now, the diagonal of the square will be $\sqrt{(\sqrt d)^2+(\sqrt d)^2} = \sqrt{2d}$ 

Now, $\sqrt{2d}$ is the perimeter of a circle. 

∴ Radius of the circle will be $\dfrac{\sqrt{2d}}{2}$

We know $\color{black}{\textbf{Area of the circle}}$ is $\pi r^2$, where, $r = \text{radius of the circle}$

$ = \dfrac{22}{7} \times \dfrac{\sqrt{2d}}{2} \times \dfrac{\sqrt{2d}}{2}$

$= \dfrac{22}{7} \times \dfrac{2d}{2 \times 2} $

$= \dfrac{22}{7} \times \dfrac{d}{2}$

$=\color{Black}{ \pi . \dfrac{d}{2}}$

$\text{Option (D)}$

Comments

$\sqrt{2d}$ is the *diameter of the circle.

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yes,$\sqrt{2d}$ is the diameter of circle not perimeter

Answer 2

Area of square = d.
therefore, side of square = √d

Now using Pythagoras theorem we can find the diagonal of the square.
i.e  $h^{2}$ = $p^{2}$ + $b^{2}$, here both p and b are √d
therefore diagonal (h) =  √2d

It is given that diagonal of the square is the diameter of the circle.
therefore radius = (√2d)/2

Area of the circle = π$r^{2}$
i.e π x (√2d)/2 x (√2d)/2
i.e π x (2d)/4
i.e πd/2

Hence, the correct answer is option d.

Answer 3

Answer is option D