Question

A sphere of radius $r$, inside a cube touches each one of the six sides of the cube. What is the volume of the cube in terms of $r$ ?

• A. $8r^3$
• B. $2r^3$
• C. $4r^3$
• D. $\frac{4}{3}\pi r^3$

Answer 1

The correct option is A $8r^3$

Given that

A sphere of radius ${}^{\prime }{r}^{\prime }$ inside a cube touches each one of the six sides.

Let ${}^{\prime }{s}^{\prime }$ be the side of the cube

To find the volume of the cube,

we need to find the side of the cube in terms of ${}^{\prime }{r}^{\prime }$.

The center of the cube is the same as the center of given sphere.

As the sphere touches the side, the distance between the center of the sphere(or cube) and the side of the cube is the $radius{(}^{\prime }{r}^{\prime })$ of sphere.

We know that,

If ${}^{\prime }{s}^{\prime }$ is the side of the cube,

The distance between the center of the cube and the side of the cube $=$ $\frac{s}{2}$

Here,

$r$ $=$ $\frac{s}{2}$ $\left(fromabove\right)$

$s$ $=$ $2r$

Side of the cube in terms of 'r' $=$ $2r$

Volume of the cube $=$ ${\left(sideofthecube\right)}^3$

$=$ $s^3$

$=$ ${\left(2r\right)}^3$

$=$ $8$$r^3$

Therefore, Volume of the cube is ${}^{\prime }$$8$$r^3$${}^{\prime }$.