Multiple choice

A sphere of radius $r$ lies inside a cube and touches each of the six sides of the cube. Calculate the volume of the cube in terms of $r$.

  1. $\displaystyle { r }^{ 3 }$
  2. $\displaystyle 2{ r }^{ 3 }$
  3. $\displaystyle 4{ r }^{ 3 }$
  4. $\displaystyle \frac { 4 }{ 3 } \pi { r }^{ 3 }$
  5. $\displaystyle 8{ r }^{ 3 }$
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E Correct answer
Explanation

If a sphere of radius r is inscribed in a cube, the side length of the cube is equal to the diameter of the sphere, which is 2r. The volume of the cube is (side)^3 = (2r)^3 = 8r^3.

AI explanation

When a sphere of radius r touches all six sides of a cube, the side length of the cube is equal to the sphere's diameter, which is 2r. The volume of a cube is calculated as side^3. Substituting the side length gives (2r)^3 = 8r^3.

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