Question:medium

What is the area of the square, if four vertices lie on the circumference of a circle where the area of the circle is four times its diameter in magnitude?

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Use the area equals four times diameter condition to find the radius, then relate the square's diagonal to the circle's diameter.
Updated On: Jul 21, 2026
  • \( \frac{8}{\pi^2} \) sq. units
  • \( \frac{16}{\pi^2} \) sq. units
  • \( \frac{32}{\pi^2} \) sq. units
  • \( \frac{128}{\pi^2} \) sq. units
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The Correct Option is D

Solution and Explanation

Step 1: Find the circle's radius from the given condition.
Area of circle equals 4 times diameter, so $\pi r^2 = 4(2r) = 8r$, giving $r = \frac{8}{\pi}$ after dividing by r.

Step 2: Find the side of the inscribed square using its diagonal.
The square's diagonal equals the circle's diameter, $2r$.
If the side of the square is $a$, then by Pythagoras $a^2+a^2 = (2r)^2$, so $2a^2 = 4r^2$, giving $a^2 = 2r^2$.

Step 3: Compute $r^2$ and then $a^2$.
$r = \frac{8}{\pi}$, so $r^2 = \frac{64}{\pi^2}$.
$a^2 = 2 \times \frac{64}{\pi^2} = \frac{128}{\pi^2}$.

Step 4: State the area of the square.
The area of a square equals $a^2$ itself, so the area is $\frac{128}{\pi^2}$ square units.

Final Answer:
The side length approach also gives an area of $\frac{128}{\pi^2}$ square units. \[ \boxed{\frac{128}{\pi^2} \text{ sq. units}} \]
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