Let the radius of the sphere be ‘R’ units

And the cube which will fit inside it be of edge ‘a’ units

Explanation: The longest diagonal of the cube that will fit inside the sphere will be the diameter of the of the sphere.

∴ The longest diagonal of cube = the diameter of the sphere

Consider ΔBCD, ∠BDC = 90°

BD = CD = a units (as they are the edges of cube)

⇒ BC² = a² + a² (putting value of BD and CD)

⇒BC² = 2a²

⇒BC = √(2a²)

∴ BC = a√2 units →eqn1

Now consider ΔABC, ∠ABC = 90°

Here, AB = a units and BC = a√2 units

⇒AC² = a² + (a√2)² (putting values of AB and BC)

⇒ AC² = a² + 2a²

⇒AC² = 3a²

⇒AC = √(3a²)

∴ AC = a√3 units

∴ Diameter of sphere = D = a√3 units

And we know, D = 2 × R

⇒ R = D/2 (put value of D )

Also, Volume of a sphere→eqn2

Put value of R in eqn2

∴ Volume of sphere = πa² cubic units → eqn3

Volume of cube = (edge)³

∴ Volume of cube = a³ cubic units →eqn4

Ratio of volume of cube to that of sphere

(putting values from eqn3 and eqn4)

⇒Ratio of volume of cube to that of sphere

Ratio of volume of cube to that of sphere is a:π