That proof is not possible, because the volume should come πr³/12 cubic cm, not 1/6.
Volume of original cube = r³ cubic cm.
Cone is of largest size.
=> height of cone = side of cube = r cm
=> diameter of cone = side of cube = r cm
=> radius of cone = diameter/2 = r/2 cm
∴ Volume of cone
= πr³/12 cubic cm.