Question

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?

• A. \( \frac{8}{\pi^2} \) sq. units
• B. \( \frac{16}{\pi^2} \) sq. units
• C. \( \frac{32}{\pi^2} \) sq. units
• D. \( \frac{128}{\pi^2} \) sq. units

Hint:

Use the area equals four times diameter condition to find the radius, then relate the square's diagonal to the circle's diameter.

Answer 1

The Correct Option is D

Step 1: Use the given condition to find the radius.
Let the radius of the circle be r, so its diameter is \( 2r \).
The area of the circle equals four times its diameter in magnitude, so \( \pi r^2 = 4(2r) = 8r \).
Dividing both sides by r gives \( \pi r = 8 \), so \( r = \frac{8}{\pi} \).

Step 2: Relate the square's diagonal to the circle's diameter.
Since all four vertices of the square lie on the circle, the square is inscribed in the circle.
The diagonal of the inscribed square equals the diameter of the circle, so diagonal \( = 2r \).

Step 3: Find the area of the square from its diagonal.
For a square with diagonal d, the area is \( \frac{d^2}{2} \).
Area \( = \frac{(2r)^2}{2} = \frac{4r^2}{2} = 2r^2 \).

Step 4: Substitute the value of r.
\( r = \frac{8}{\pi} \), so \( r^2 = \frac{64}{\pi^2} \).
Area \( = 2 \times \frac{64}{\pi^2} = \frac{128}{\pi^2} \) sq. units.

Final Answer:
The square inscribed in the circle has area \( \frac{128}{\pi^2} \) square units. \[ \boxed{\frac{128}{\pi^2} \text{ sq. units}} \]