A cube of side a is inscribed in a sphere of radius R. A plane parallel to one face of the cube is drawn at a distance frac R sqrt 3 from the center of the sphere. What is the area of the circular cross-section?

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EEJ MAIN Aptitude QUESTION #10991

Question 1

A cube of side $a$ is inscribed in a sphere of radius $R$. A plane parallel to one face of the cube is drawn at a distance $\frac{R}{\sqrt{3}}$ from the center of the sphere. What is the area of the circular cross-section?

$\pi R^2$
$\frac{2\pi R^2}{3}$✔️
$\frac{\pi R^2}{2}$
$\frac{4\pi R^2}{9}$

Correct Answer Explanation

The sphere has radius $R$. The plane is at distance $d = \frac{R}{\sqrt{3}}$ from center. Cross-section radius: $r = \sqrt{R^2 - d^2} = \sqrt{R^2 - \frac{R^2}{3}} = R\sqrt{\frac{2}{3}}$. Area = $\pi r^2 = \frac{2\pi R^2}{3}$.

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