Answer
The side-length of the cube is $\sqrt2$.
Work Step by Step
We know that one face of the cube is contained in the circular base of the cone.
We also know that the perpendicular bisector of a chord passes from the centre of the circle.
Since the side of the square face of the cube is a chord on the circular base and perpendicular bisector of the side of a square passes from the intersection of its diagonals.
The diagonals of the square face of the cube intersect at the centre of the circular base of the cone.
Thus, the diagonal of the square face is the diameter of the circular base.
Which is $2$.
Now using the Pythagorean theorem.
We get, $2^2=a^2+a^2$, where $a$ is the side-length.
$\implies 4=2a^2$
$\implies a^2=2$
$\implies a=\sqrt2$
Hence, the side-length of the cube is $\sqrt2$.
