Question

Determine the ratio of the length of the longest rod that can be fitted in a sphere to the length of the longest rod that can be fitted in a cube, given that the sphere exactly fits inside the cube.

• A. $\sqrt{3}:1$
• B. $1:\sqrt{3}$
• C. 1 : 1
• D. 2 : 1

Answer 1

The correct option is B

$1:\sqrt{3}$

Let the side of the cube be $x$.
$\therefore$ Diameter of sphere$=x$.
[The sphere fits exactly into the cube]

Length of the longest rod that can be fit into the cube
= main diagonal of the cube across the opposite vertices
=$\sqrt{3}x$

Length of the longest rod that can be fit into a sphere is the diameter i.e. $x$.

$\therefore$ Ratio of length of the longest rod in sphere to the length of the longest rod in cube
$=\frac{x}{\sqrt{3}x}$

$=\frac{1}{\sqrt{3}}$