Review question

# If a sphere contains this cuboid, what's the largest cube it can contain? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9973

## Solution

A cuboid has sides of lengths $22$, $2$ and $10$. It is contained in a sphere of the smallest possible radius. What is the side-length of the largest cube that will fit inside the same sphere?

(A) $10 \quad$ (B) $11 \quad$ (C) $12 \quad$ (D) $13 \quad$ (E) $14$

Any cuboid fitting snugly inside a sphere will have all $8$ vertices touching the sphere, and by symmetry, the diagonal of the cuboid will be a diameter of the sphere.

Let the diagonal of the given cuboid be $d$. Then by Pythagoras’ theorem we have $d^2=22^2+2^2+10^2 = 588.$

If the cube has side length $x$ then we also have $d^2=x^2+x^2+x^2$ and so $3x^2=588 \quad\implies\quad x=14.$