Question

Determine the ratio of the volume of a cube to that of the sphere which will exactly fit inside the cube.

Answer 1

Problem Statement: Given a cube and an inscribed sphere, calculate the ratio of the cube's volume to the sphere's volume.

Cube Volume Formula: The volume of a cube is \( V_{\text{cube}} = a^3 \), where \( a \) is its side length.

Sphere Volume Formula: The volume of a sphere is \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \), where \( r \) is its radius.

Geometric Relation: For a sphere inscribed in a cube, the sphere's diameter equals the cube's side length: \( 2r = a \), which implies \( r = \frac{a}{2} \).

Sphere Volume in Terms of Cube Side: Substitute \( r = \frac{a}{2} \) into the sphere volume formula: \( V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{a}{2} \right)^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{\pi a^3}{6} \).

Volume Ratio Calculation: The ratio of the cube's volume to the sphere's volume is \( \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{a^3}{\frac{\pi a^3}{6}} = \frac{6}{\pi} \).

Result: The ratio of the volume of the cube to the volume of the inscribed sphere is \( \frac{6}{\pi} \).