If a sphere is inscribed in a cube
of volume eight cubic centimeters, what is the volume of the sphere?
The key to this question is being
able to relate the dimensions of the cube to the sphere. Since the sphere is inscribed in
the cube, this means that the sphere is touching each face of the cube without any
gaps. As such, the diameter 𝑑 of the
sphere is equal to the length 𝑙 of the cube. Alternatively, the radius 𝑟 is
half of the length of the cube.
Next, we recall that the volume of
any cube is equal to its side length cubed. And in this question, we’re told
that this volume is eight cubic centimeters. This means that 𝑙 cubed is equal
to eight. We can then cube root both sides
such that 𝑙 is equal to two. The side length of the cube is
therefore equal to two centimeters.
We have already established that
the radius of the sphere is half of this. And this is therefore equal to one
centimeter. We can calculate the volume of any
sphere when we know its radius. The volume is equal to four-thirds
𝜋𝑟 cubed. If we let the volume of our sphere
be 𝑉, we have 𝑉 is equal to four-thirds 𝜋 multiplied by one cubed. And this is simply equal to
four-thirds 𝜋. We can therefore conclude that if
the volume of the cube is eight cubic centimeters and the sphere is inscribed in the
cube, then its volume is four-thirds 𝜋 cubic centimeters.