A perfect cube with a side length of 8 cm has the largest possible sphere fitted inside it. What is the volume of the empty space within the cube?

Correct — A. The largest sphere inside a cube touches all six faces, so its diameter = side = 8 cm.

Radius of sphere = 8 ÷ 2 = 4 cm
Volume of cube = 8³ = 512 cm³
Volume of sphere = 4⁄3 π × 4³ = 4⁄3 π × 64 = 256π⁄3 cm³
Empty space = cube − sphere = 512 − 256π⁄3 cm³ → option (a)

Why the others are wrong

• (b)
  512 − 64π⁄3 cm³ — 64π⁄3 is 4⁄3 π × 4², the radius squared instead of cubed. The sphere's volume needs r³ = 64, which gives 256π⁄3.
• (c)
  64 − 256π⁄3 cm³ — 64 is 4³, the sphere's radius cubed, not the cube's volume 8³ = 512. And 64 − 256π⁄3 is negative, since 256π⁄3 ≈ 268.
• (d)
  256π⁄3 cm³ — 256π⁄3 is the sphere's own volume, the space the ball fills. The question asks for the space left empty, so subtract it from 512.

A sphere inscribed in a cube touches the centre of each face, so the sphere's diameter equals the cube's edge: r = a⁄2.

Empty space = a³ − 4⁄3 π (a⁄2)³ = a³ − πa³⁄6 = a³(1 − π⁄6).

So the sphere always fills π⁄6 ≈ 52.4% of the cube, and about 47.6% stays empty, whatever the side.

Key facts

• The largest sphere in a cube of side a has radius a⁄2.
• Volume of a sphere = 4⁄3 π r³, and for r = 4, r³ = 64.
• An inscribed sphere occupies π⁄6 of the cube's volume, about 52.4%.

Common traps

• Taking the sphere's radius as 8 cm, the full side, instead of 4 cm.
• Using r² in the sphere's volume, which gives 64π⁄3.

No directly related past PYQ was found.