Multiple choice

Area of the cyclic square is 256 cm $^{2}$, then what is the area of the circle inscribing the square

  1. 64$\pi cm^{2}$
  2. 118$\pi cm^{2}$
  3. 136$\pi cm^{2}$
  4. 128$\pi cm^{2}$
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D Correct answer
Explanation

Area of square = s^2 = 256, so side s = 16. The diagonal of the square is the diameter of the circumscribing circle. Diagonal d = s*sqrt(2) = 16*sqrt(2). Radius r = d/2 = 8*sqrt(2). Area of circle = pi * r^2 = pi * (8*sqrt(2))^2 = pi * 64 * 2 = 128*pi.

AI explanation

The area of the square is 256 cm$^2$, so its side length $s$ is $\sqrt{256} = 16$ cm. The diagonal of the square is the diameter of the circumscribed circle, calculated as $d = s\sqrt{2} = 16\sqrt{2}$ cm. The radius of the circle is half the diameter, so $r = 8\sqrt{2}$ cm. The area of the circle is $\pi r^2 = \pi (8\sqrt{2})^2 = 128\pi$ cm$^2$.

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