Geometry

Solids

Medium

A sphere fits exactly in a hollow cube. Find the ratio of the volumes of the cube and the sphere

$\pi : 6$

1 : 6

$6 : \pi$

6 : 1

✅ Correct Option: 3

When a sphere fits exactly inside a cube, it touches all 6 inner faces of the cube. This means the sphere's diameter equals the cube's side length.

If the side of the cube $= a$ , then the radius of the sphere $= \dfrac{a}{2}$

Volume of the cube $= a^3$

Volume of the sphere $= \dfrac{4}{3}\pi r^3$

$= \dfrac{4}{3} \times \pi \times \left(\dfrac{a}{2}\right)^3$

$= \dfrac{4}{3} \times \pi \times \dfrac{a^3}{8}$

$= \dfrac{4\pi a^3}{24}$

$= \dfrac{\pi a^3}{6}$

$V_{\text{cube}} : V_{\text{sphere}} = a^3 : \dfrac{\pi a^3}{6}$

Cancelling $a^3$ from both sides:

$= 1 : \dfrac{\pi}{6}$

$= 6 : \pi$

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