$\"\"$

A cube is inscribe in a cone such that one of the face of cube touches the base of the cone. \ \

Height of the cone is 3 units \ \

Radius of the circular base is 1 unit. \ \

Let us consider $\"\"$ be the angle. \ \

$\"\"$ \ \

$\"\"$

From the Larger triangle made by the cone, $\"\"$ . \ \

From the Smaller triangle made by the cube, $\"\"$ .

$\"\"$

We know that diameter of the base of the cone is equal to the diagonal of the cube. \ \

radius is half of the diagonal of the cube. \ \

$\"\"$ . \ \

Then $\"\"$ .

$\"\"$

$\"\"$ .

Therefore side length of cube is $\"\"$ units. \ \

$\"\"$ \ \

Side length of cube is $\"\"$ units.